‘a lot’ can be used as an adverb
I go to the gym a lot [very often].
‘a lot’ can also be used as a quantifier
There are a lot [many, quantifier] of children.

They must have received a lot of private English education. [In this sentence, ‘a lot’ is a quantifier. The amount of education they received is large].

They must have received private English education a lot. [In this sentence, ‘a lot’ is an adverb. They received English education frequently. This sentence is not really wrong, but it is unnatural. It would be better to change the verb ‘received’ to ‘been to’ or ‘attended’, and ‘education’ to ‘classes’].

They must have attended private English classes a lot.