How can one maintain the ritual purity (tahara) of an infant during Umrah, and is it permissible to cut their hair?

Here is how to perform the 'Umrah for a child:

It is important to maintain the cleanliness of the child's clothes and body, especially during the circumambulation (Tawaf), because ritual purity (taharah) is a condition for the validity of Tawaf. The guardian should ensure the child is wrapped securely so that impurities do not spread to their clothes.

As for cutting the child's hair after entering the state of ihram, it is not permissible because it is one of the prohibitions of ihram. Rather, their hair should be cut after performing the Tawaf and the Sa'y (running between Safa and Marwah).

If the child is not yet discerning (ghayr mumayyiz), their guardian should make the intention of ihram on their behalf, remove any sewn garments from them, and recite the Talbiyah for them. The child then enters the state of ihram and is subject to the same prohibitions as an adult in ihram. Both the guardian and the child should have clean clothes and bodies during Tawaf. There are two opinions regarding the condition of purity for a non-discerning child during Tawaf; the most authoritative view according to the Shafi'i school is that it is a condition.