I'm sorry, I'm really not trying to be picky here, but I really don't understand your question (thanks for your understanding, by the way)write the correct form:
we must do a lot of homework at this school
is the mistake in ( a lot)
can we say
we must do lots of ......etc
I don't often disagree with you Mole, but I'm not with you on We must, as opposed to We have to. I think you might say must if you meant that to keep up we must do lots of homework, while we have to suggests more to me that the school authorities make one do it: I'm not suggesting that the words have these exclusive meanings, but that I think We must is perfectly correct in some circumstances.No, "a lot" here is fine, however, "We have to do a lot of homework" is probably what you mean. "Must" in this sentence, tends to suggests the usage of must as in "it must be the case", and I think you mean "we are obliged to do" a lot of homework.
"Lot's of" is also fine. "We have to do lots of homework".