glaoch is a noun. cuir is a verb. Where English has an infinitive verb with a subject, Irish puts the subject (noun) before the verbal noun.
An bhfuil tú chun an múinteoir a fhreagairt?
Nílim chun an bhille sin a íoc
Chuaigh mé inniu chun m'fhuil a thabhairt
Nílim in ann é sin a fháil
An bhfuil tú in ann iad a fheiceáil?
Dúirt mé leat gan é sin a chur sa mhála
The Duolingo notes for this level only talk about the object of a sentence coming after a verbal noun (ag + verbal noun + object in genitive form), hence my confusion. Is it 'object + a + lenited verbal noun' here because the verbal noun functions as --in English-- "to do something" rather than "doing something"? (I don't know the grammatical terms for this in English, sorry)
It's helpful to make some technical distinction in this case because in English "doing something" can be either a gerund ("I love doing something") or a progressive verb ("I am doing something right now"), and Irish would translate these differently.
My impression is that the "ag + verbal noun + object" construction is used for the progressive and "object + a + verbal noun" is used for either gerunds or infinitives ("I love doing something" OR "I love to do something"), but I'm kind of out of my depth at this point :P