Handbook of ROCOR Liturgical Practice:

When does a Hieromonk remove his klobuk (hat with veil) during the Divine Liturgy?

In general, the rules applying to when heads are covered and uncovered, as they relate to Priests with kamilavkas (see a different entry in this Handbook) apply also to monastics. However, Hieromonks remove their klobuks after entering the Holy Alter following the Little Entrance and serve without them for the remainder of the Liturgy (to the Prayer of the Amvon).

This means that the Hieromonk’s klobuk is removed significantly earlier in the service than the kamilavka of a Priest.

When a monastic is of the rank of Abbot or higher, then the klobuk remains on until the Cherubic Hymn (and in some cases is replaced with the kamilavka for Abbots in particular).