Just seeing the word rúnaí along side "secretary" here made me realise, for the first time, their parallel etymologies. "Secretary" is from "secret" (how did I never notice that before?), because it refers to a confidential advisor, and similarly rúnaí is from rún secret.
"Cá bhfuil a rúnaí?" is what I heard. It also makes sense, so it should be accepted as "Where is his/her secretary". Siúil, siúil, siúil a Rooney; déan deifir, brostaigh ort - you're needed on the pitch!
'an' is pronounced 'a' before consonants in normal speech. For words beginning with r-, l-, n-, etc., this would make "the" identical with "his", "her", and "their", if the speaker doesn't make the effort to say the "n".