"I am cleaning my clothes."
Translation:Táim ag glanadh mo chuid éadaí.
You're right, of course, I wasn't paying any attention to the verbal noun. But, from Gramadach na Gaeilge
The form of the genitive is not used when (...) when two nouns in the genitive directly follow one another, to avoid a double genitive. Instead the first is lenited in the nominative and only the second is in the genitive. This is the so-called "functional genitive", the first noun is "functional" in genitive relation, but keeps the nominative form and is lenited.
So cuid isn't in the genitive form because éadaí is in the genitive. And mo would cause lenition anyway, but as far as I know, the mutation from the possessive adjective would take precedence, so you would still say ag glanadh ár gcuid éadaí.