Kohanim are not allowed to marry a convert. However, if such a case happens (where there is a fully valid conversion), what is the status of the child? On one hand, they would be chalal, but would that offspring be able to inherit the Levite tribe of their father?
Various shiurim and tapes have explained that a chalal is treated a a Yisroel and not a Levite. For example Rabbi Kaganoff gives the example of the son of a kohen who had married a divorcee had to perform a Pidyon Haben on his first born child.
The shiurim that I have seen explain the reasoning behind this, but I do not have the explicit citations right now.
Rambam does say
Yet if a chalal went ahead and offered a korban, his offerings are accepted after the fact (Maimonides, Bi'at Mikdash 6:10).
However, this is different and normally he is not allowed to even join the Leviim to sing or perform the Avodah of Levi in the Bais Hamikdash.