Imperial Russia considered all its subjects to fall within a social class or estate (petty burgher, noble, privately held serf, native, etc.) Most of these categories are hereditary, but a couple hinge on the career of the individual and would seem to expire with the holder rather than be carried on by any children. What would then be the social class of the child of a clergyman or an exile?
There is a very detailed discussion of this issue on pages 281-284 of
In short: The status of children of clergy depended on the type of clergy, on clergy's education, on the region of residence, etc. For instance, the rules were different for a child of a "white" (nothing to do with race, just means "not a monk") orthodox priest and for a child of a rabbi.
Subject to some limitations, in the case of Russian orthodox priests and of Armenian, Muslim (in the Caucasus region) or Lutheran Protestant clergy (in the Baltic provinces) who were not of the "noble" origin, their children would acquire the status of a "Citizen of Honor" (Почётный гражданин). You can find more about this on this wikipedia page in French.
Edit. Here is a brief description of privileges that citizens of honor had in Tsarist Russia:
Freedom from conscription.
Freedom from the poll tax.
Freedom from corporal punishment.
Right to be elected in municipal public offices.
Lastly, the rank of a citizen of honor in pre-revolutionary Russia has nothing to do with the title of a honorary citizen that can be awarded by a municipality or a country as a sign of recognition for some special services. (E.g. Raoul Wallenberg was posthumously awarded the title of a honorary citizen of Canada, Hungary, Australia and Israel.)