Short answer
No. A man's head is not awrah, so covering it is not obligatory, in prayer or outside it. Wearing a cap (kufi, topi) or a turban is a sunnah of habit and a long-standing Muslim custom. In the Hanafi school, praying bareheaded out of laziness is makruh, while doing so out of humility is fine. The four schools agree that it is not required.
A man's head is not awrah, so covering it is not required. Wearing a cap or turban is a sunnah of habit and good manners. Praying bareheaded is makruh when it comes from laziness or carelessness, but not when done out of humility.
The four schools agree on this.
Explanation
The Prophet ﷺ usually covered his head with a turban or a cap, and the Companions did the same. Over the centuries Muslims came to see a head covering as part of dignified dress, especially for prayer, and many communities still keep it.
Not an obligation. A man's awrah is from the navel to the knees, so his head does not have to be covered and his prayer is valid without a cap. See the awrah guide. Reading the Quran, making dua or entering the mosque with your head uncovered is allowed.
In prayer. Allah tells us to take our adornment at every place of prayer. The Hanafi books say praying bareheaded out of laziness, or because a cap feels like a burden, is makruh. If a man uncovers his head deliberately out of humility before Allah, there is no harm. If your cap falls off during prayer, putting it back with one hand is better, as long as it does not take much movement.
Which cap. A kufi, topi, taqiyah, crocheted cap, Turkish or Malay cap, a ghutra with agal, or a turban are all fine. None is required, and none is more Islamic than another. For the turban, see is wearing a turban sunnah.
Why some men always wear one. Many wear a cap as a sign of Muslim identity and modesty, or out of love for the dress of the righteous. That is a good intention. Do not look down on brothers who do not wear one.
The evidence
Quran, Surah Al-A'raaf 7:31
۞ يَٰبَنِىٓ ءَادَمَ خُذُوا۟ زِينَتَكُمْ عِندَ كُلِّ مَسْجِدٍۢ وَكُلُوا۟ وَٱشْرَبُوا۟ وَلَا تُسْرِفُوٓا۟ ۚ إِنَّهُۥ لَا يُحِبُّ ٱلْمُسْرِفِينَ ﴿٣١﴾
O children of Adam, take your adornment at every masjid, and eat and drink, but be not excessive. Indeed, He likes not those who commit excess.
Dressing well for prayer is encouraged, which is why the Hanafis dislike praying bareheaded out of laziness.
Sahih Muslim 1358
Sahih... دخل يوم فتح مكة وعليه عمامة سوداء ...
"He entered Makkah on the day of the conquest wearing a black turban."
The Prophet covered his head as a habit.
Where the ruling is recorded
- al-Durr al-Mukhtar with Radd al-Muhtar, Kitab al-Salah, chapter on what invalidates the prayer and what is disliked in it. Praying bareheaded out of laziness is makruh, there is no harm if it is out of humility, and replacing a fallen cap is better.
- al-Hidaya, Kitab al-Salah, chapter on the conditions of prayer. A man's awrah is from the navel to the knee.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.