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What does one do in a situation that he needs to walk and does not have a head covering?
Example: What does one do if his yarmulke flew off and is found more than 4 cubits from him?
If one is able to cover his head with his hand he is to do so, as explained above in D. If he is unable to do so [such as when he is carrying things in his hands] then some Poskim rule he should walk less than four cubits at a time. This means that he should walk, stop, walk, stop, until he reaches his destination.
What is one to do if he needs to say a blessing and does not have a head covering?
One is to have another person place his hand over the head or the person himself is to cover his head with his sleeve. Some rule that in a time of need one may cover his head with his own hand [if he does not have a sleeve that can reach his head].
May one work a job which demands him to leave his head uncovered?
No. However there are Poskim that are lenient in a case of great loss. One should strengthen his trust in Hashem to find a job without such conditions.
If one Davened or recited a blessing without a Yarmulke is he to repeat the blessing or prayer?
If one prayed or recited a blessing without a head covering, then if he did so unknowingly, many Poskim rule he is not required to repeat the blessing or prayer. If however he knowingly prayed without a head covering he must repeat the Davening. Some Poskim however lean to rule that one must repeat the prayer in all cases even if he unknowingly prayed without his head covering. [All the above is with regards to prayer, however regarding a blessing seemingly according to all he would not be required to repeat it.]
Must one cover his head during a haircut?
No. However some are particular not to completely uncover the head during a haircut.
Must one wear a Yarmulke in a bathhouse or Mikveh?
No. One may remove it in the dressing room and walk bareheaded to the shower and Mikvah. Some Poskim rule there is no need to be stringent even as an act of piety. Others rule there is an act of piety involved in wearing a Yarmulke when walking even in a bathhouse. Thus those that are meticulous should cover the head with their hand even in the shower room, when walking four Amos. Some are even particular to wear the Yarmulke when they are in the water and hence never remove it. Upon returning from the Mikvah into the dressing room, one is to wear his Yarmulke right away and prior to wearing any other clothing.
Must one wear a Yarmulke while in the ocean or swimming pool?
No. However one is to put his Yarmulke on as soon as he leaves the water. Some are stringent to wear a Yarmulke even when they are in the water.
May one offer food to Jews who do not wear a head covering?
Yes. However one is to instruct them to say a blessing beforehand and it is also proper to propose them to cover the head. If doing so will offend the person then it is not necessary to mention this when offering them food.
Must women cover their head when reciting a blessing?
Married women must cover their heads upon saying a blessing or prayer. Single girls are not required to cover their heads even during prayer.
Does an old Yarmulkeh require Geniza?
No. Nevertheless it is proper to wrap it in a bag and place it in the garbage rather than placing it directly in together with all the filth.
 Ketzos Hashulchan 3/8
The Reason: Being that walking four cubits without a head covering is more severe than walking less than four cubits, therefore one should follow the above method. [see also Chapter 3 Halacha 3 regarding one who does not have water near his bed upon awakening]. Vetzaruch Iyun Gadol as by doing so one is delaying his head from being covered and thus transgressing Tznius and “In their statues they shall not follow”. Hence it seems Pashut that according to Admur in Basra if one is outside he should certainly go home as soon as possible and not be without a head covering. Vetzaruch Iyun.
 Ketzos Hashulchan 3/8
Being that walking four cubits straight without a head covering is more severe then walking less then this amount, therefore
 Admur 91/4; Piskeiy Teshuvos 2/10
 M”B 2/12
 Elya Raba 91; M”B 2/12 relying on opinion of Rashal; Shulchan Hatahor 2/1; 91/2; Vetzaruch Iyun according to Admur if he would allow one to be lenient in a case of need like the Rashal, just like he rules regarding another person’s hand that even initially one may be lenient like the Rashal.
 Beir Moshe 8/40; As one transgresses “Chukos Hagoyim”. See also Toras Yekusiel 12; Minchas Yitzchak 3/33; Piskeiy Teshuvos 2 footnote 57
 Igros Moshe 4/2; 5/11; based on those that rule it is a mere custom to wear a head covering and hence in a case of loss one may be lenient against the custom. This would not apply according to Admur.
 Beir Moshe ibid; See Igros Kodesh 3 p. 372
 Piskeiy Teshuvos 2/10; Igros Moshe 4/40
 Yeshuos Moshe 3/15; Halichos Shlomo 2/16; Oar Letziyon 2/7-13; Yabia Omer 9/1; Piskeiy Teshuvos 2/10; 91/2
 As his Davening is an abomination, just like one who prayed near feces. [ibid]
 Igros Moshe 4/40 as perhaps the mere act of prayer without a head covering is an abomination. [ibid]
 As even one who knowingly said a blessing near feces is not to repeat the blessing due to doubt. [185/5]
 M”B 2/11
 See Piskeiy Teshuvos 2 footnote 84; Biur Halacha 2
 See Shaareiy Teshuvah 2/3; Kaf Hachaim 2/17; Piskeiy Teshuvos 2/12
 Bechor Shur brought in Shaareiy Teshuvah ibid; Birkeiy Yosef 2/2
 Ben Ish Chaiy Vayishlach 16 [Wear it until reach the shower room]; Custom of meticulous Jews, brought in Bechor Shur ibid and so he [Rav Gedalya Shur] testifies to be his personal custom; Shulchan Hatahor 2/3
 Kaf Hachaim ibid and he concludes that so is the custom amongst some people.
 Shulchan Hatahor Tznius; See Minchas Yitzchak 4/60
A story with Reb Hillel: It is told of Rav Hillel Paritcher that he would wear a Yarmulke even while immersing in the Mikvah. The Yarmulke would float on the water and when he would lift his head from under the water he would do so in a way that the Yarmulke went on his head immediately upon exiting. It once occurred that he could not find his Yarmulke and he hence did not lift his head from the water for quite some time. Miraculously he survived the ordeal. [To note that Reb Hillel was known for his excessive stringencies which was not meant as a directive for the public to follow.]
 See Halacha 6B in Q&A
 Piskeiy Teshuvos 4/12 footnote 82
 Sheivet Halevy 8/47; Piskeiy Teshuvos 2/10
 It is permitted from the letter of the law to give them food without mention about a head covering however it is best to mention it. However regarding a blessing one must tell them to say a blessing as explained in 169/2. [Sheivet Halevy ibid]
 See Piskeiy Teshuvos 169/3
 See Yabia Omer 6/15; Tefilah Kehilchasa 7/74; Piskeiy Teshuvos 91/2
 See Piskeiy Teshuvos 21/2 footnote 42
See Shu”t Igros Moshe Orach Chaim Vol. 4 Siman 40:14 where he writes that Tefilah davened with an uncovered head is not valid B’dieved. Some rabbonim want to deduce from there that Rav Moshe Zatzal would Pasken the same thing for blessings.
See Avnei Yushfei 5:33:3 that a blessing recited without a head covering is not valid.
Regarding prayer with an uncovered head – See Eishei Yisroel 10-8 ftnt. 33. Where he paskens like R’ Moshe, Igros Moshe O:C 4-40 ( 14) to repeat his prayer that he said with an uncovered head, and he also brings R’ S. Z. [who is lenient not to repeat], but he still paskens like R’ Moshe. (BTW the Eishei Yisroel was written by a student of R ’S. Z. and was reviewed by R’ Y. Y. Neuwirth Zt”l, a talmid of R’ S. Z.)
It could be that there is no disagreement between R’ Moshe and R’ Shlomo Zalmen. R’ Moshe clearly states that his psak is because “in our countries,” it is the norm for the goyim to pray specifically with an uncovered head, out of respect [and therefore for a Jew to pray in such a manner is an abomination in our countries].
R’ S. Z brings a midrash that an uncovered head is not meakev the davening. That might have been fine during the time of the midrash, however R’ Moshe was writing something else.
Therefore, practically speaking, it is my humble opinion that one who prayed SE with an uncovered head is to repeat SE, at least as a voluntary prayer [making a stipulation that if he is obligated to repeat, he is doing so, and if not, he is merely reciting a voluntary prayer], and even on Shabbos and Yom Tov [when it is forbidden to recite a voluntary prayer], he is to either hear the prayer from the chazzan or even to repeat it himself.