You really do not need to empty and deep clean the cage monthly. Hamsters rely on their cage having their scent to feel secure and only soiled bedding needs removing. I would not remove the hoarded food either because having that helps them to feel safe.
I think us humans like everything to look clean and tidy but there is no need to with hamsters.
I find these videos on hamsters so helpful so I will put a link to one just in case you would like to see too.
How I spot clean my cages! - YouTube