"I cannot buy new clothes."
Translation:Ní féidir liom éadaí nua a cheannach.
The 'a' with a fada in an earlier construction meant 'at its'. What does this 'a' without a fada literally mean?
This a is a preposition that literally means “to”, in the English infinitive particle sense of ”to buy”.
There's something very folk-songy about this construction. He went down to the market / New clothes for to buy...
No - the "infinitive with an object" construction is always object + a + Verbal-Noun.
What you were trying to get to was probably something like Ní féidir liom ceannaigh éadaí nua, but that that's not how this type of sentence is constructed in Irish, even though in English you can switch between a non-infinitive construction ("I can't buy clothes") or an infinitive construction ("I'm not able to buy clothes" or "It's not possible for me to buy clothes"), you need to stick with the infinitive construction in Irish (with the object before the verbal noun).
You're mixing up two different structures. "buying them" is á gceannach, "tó buy them" is iad a cheannach.
The first structure, using á with a fada, is derived from the possessive adjective a ("his", "hers", "their"), and uses lenition or eclipsis to indicate which a is intended.
In the second structure, the a is not a possessive adjective, and it always lenites. The pronoun iad, is explicitly mentioned, and you don't need a mutation to differentiate between "him", "her" and "them".
But more to the point, this exercise isn't uses the pronoun "them", it's using the noun "clothes", and while "buying them", is á gceannach, "buying clothes" is ag ceannach éadaí.