If one is davening and needs to hold a child at the same time, what are the parts of davening in which this is permissible and what are the parts where this would be forbidden?
During Shemona Esrei it is not allowed (see O:CH 96-1, M:B-4). If the child is crying and you are in middle of davening, you can sign to him to try to get him to stop without talking out, but if that doesn’t work and his crying is disturbing you, you can go somewhere that the child will not disturb you (Sharei Teshuva 104-1). If need be, you can also walk over to the child and quiet him without talking out (Tefila Khilchos 12 footnote 200 in the name of R’S. Z. Auerbach zt”l).
Regarding the other parts of davening, see M:B 96-1, who brings the Pri Megadim that we may also not hold something in our hands during Pesukei D’zimra and Shema, (See Ishei Yisroel 11 ftnt. 64 in the name of Horav C. Kanievsky shlit”a). Other poskim say that women may hold a child during Pesukei D’zimra and Shema, but not during Shemona Esrei.
These halachos do not apply to the other parts of davening.