Question: Can my child have a haircut during the three weeks or eat meat during the nine days?
Answer: The Shulchan Aruch (OC 551:14) rules that adults may not cut children’s hair during the nine days. The Mishna Berura (551:81 quoting the Elya Rabba) writes that this applies for the three weeks (between 17th Tammuz and 9th Av). While the Mishna Berura (Shaar Hatziyun 551:91) discusses whether young children can have a haircut, depending on the reason behind this prohibition, the Aruch Hashulchan (OC 551:31) and R’ Moshe Feinstein (Igros Moshe YD 1:224) allow children younger than six to get a haircut if necessary. R’ Ephraim Greenblatt (Rivevos Ephraim 1:336:2; 2:155:19) writes that one who has the minhag to give their son an upsherin shouldn’t schedule the haircut during the three weeks. Certainly, if a child has lice, one doesn’t need to wait to cut their hair (Nitei Gavriel, Bein Hametzarim 19:8).
The Magen Avraham (OC 551:31) writes that so long as children are too young to comprehend what we’re mourning about they don’t need to mourn and so can eat meat during the nine days. Nonetheless, the Mishna Berura (551:70 and Shaar Hatziyun 551:76) writes that most poskim disagree and children are not allowed to eat meat or drink wine unless it is for a mitzva (such as havadala wine) or for health reasons (See Aruch Hashulchan OC 551:26). Nonetheless, R’ Ephraim Greenblatt (Rivevos Ephraim 1:371; 2:155:22; 3:473:18) writes that one should not be strict with children, especially if they are weak.
R’ Moshe Feinstein (Igros Moshe OC 4:21:4) and R’ Ephraim Greenblatt (Rivevos Ephraim 3:338) write that children who normally don’t stay up on Friday night and eat earlier may be given meat before Shabbos.
In conclusion, when children are old enough to understand what Tisha B’av is about they should begin keeping the relevant halachos. Ideally, young children should not be given meat either.