The noun 'homework ' is uncountable,   'many' goes with countable things (many people, many cars......). ' Use 'a lot' with uncountable nouns (a lot of water...... a lot of trouble......) ........ SO I have a lot of homework to do. ........... You can say "I have many things/tasks to do." (because 'task' and 'thing' are countable). However, we usually use 'a lot' with countable nouns too, so most people would  say '  "I have a lot of things/tasks to do." Many is used less often in everyday speech, even when it is correct to use it..

I am busy now, I have (a lot of)(not many) homework to do.(corrected)this is a tricky one for esl learners!