Supplying milk, goes three-quarters of a mile to church (5)

'supplying milk' is the definition.
'milch' can be an answer for 'milk' (thesaurus). I am not certain of the 'supplying' bit.

'goes three-quarters of a mile to church' is the wordplay.
I cannot really see how this works, but
'mile' could be 'mi' (synonyms) and 'mi' is present in the answer.
'church' could be 'ch' (common abbreviation) and 'ch' is found in the answer.
A single letter 'l' remains which might be clued in a way I don't see.
This may be the basis of clue (or it may be nonsense).

Can you help me to learn more?