As Salaamu Alaikum,
Two things 1 can the person who called the Adnan lead the salaat being the only one in the mosque at the time? 2 can a brother hold his child and pray salaat? Is the salaat acceptable? JazakAllah khair
Wa Alaikum as Salaam,
Yes, a person who called the Adhan can lead the Salaah being the only one in the mosque at that time.
With respect to holding the child while praying, this is evident from the action of the Prophet (SAW). In a tradition recorded by Imam Bukhari, Abu Qatada (RA) narrates that the Prophet (SAW) was performing Salaah and was carrying/lifting/holding his granddaughter Umamah, the daughter of Zainab(RA). When he made Sajdah, he placed her on the ground, and when he stood up, he lifted her (or held her with him). (Sahih Bukhari)
Based on this tradition, the Scholars and Imams of Fiqh have generally allowed a person to hold his baby/child while performing Salaah.
Some however, have stated that only in cases of necessity it will be allowed to do so. Therefore, if there is no one to look after the baby/small child, and one has to perform Salaah, then it will be allowed to hold the child with one hand and perform Salaah if there is a need to do so. The Salaah will be acceptable and valid.
However, if there is no need to hold the child/baby or there is someone to look after the child, then it will be Makrooh (reprehensible) for one to hold the child while performing Salaah. (Umdah Al Qari Vol. 4 Pg. 303 Maktaba Rasheediya Queta Pakistan).
And Allah Knows best
Mufti Waseem Khan