Hello,
I came across this sentence in a child TV show. I got the meaning overall, they want to put a dress on her, but I do not understand how the sentence is built and I was hoping you could help me:

Caithfimid gúna a chur uirthi.

I do not understand if "caithfimid" here means "to wear" (if so why would it be "muid" instead of "sí"?) or "to want to do something".

It is not the first time I come across "a chur". I understand it means "to put", but I can't find what form it is, as most of the time I see "cuir", "cur" or "chuir" but never "a chur".

I am sorry for those questions, I am a beginner and I am trying to translate each sentence of the show to understand how the sentences were built, so don't worry if you are a bit technical, I do not mind.

Thank you very much.

caithfimíd - it should really be caithfimíd, but they have standardised on caithfimid - this means "we have to". It is future tense, and so notionally "we will have to", but it is normally found in the future, and really just means "we have to". The verb caith has many meanings, including wear, consume, etc, but "have to" is another meaning.

It is followed by the verbal noun construction. We have to go there: caithfimíd dul ann.
A verbal noun and object noun construction is formed like this: rud a dhéanamh (to do something).
Gúna a chur: a gown for to put (if you understand the early modern English meaning of "for to").

The pronunciation is gúna chur. The "a" is not pronounced after a vowel.

That was very clear, thank you very much.

Caithfimid is "we must" here.
That is recognizable by two things:
1) future tense (the sense of "must" is rendered by caith + future tense)
Of course, this could mean "we'll wear", "we'll throw", "we'll consume", as well ...
But more important:
2) an infinite verbal noun construction follows: gúna a chur uirthi, to put a dress on her
This is only possible if caithfimid means "we must"
"We'll wear to put a dress on her" makes no sense but "We must put a dress on her" does.

Caithfimid gúna. = We'll wear a dress. ("We must a dress" is senseless.)
Caithfimid gúna a chur orainn. = We must put a dress on us / We have to wear a dress.

Thank you very much for the explanation.
Now that I see it explained, I realize how obvious it was. I was confused by "caithfimid" given it was talking about clothes.

Thanks both of you!