Hi - we've had both mice and a rat once. Yes a mouse could get in a hamster cage if the bar spacing is bigger than 7mm. We put these traps down for mice. They work virtually instantly - puts the mouse to sleep within seconds and they die while asleep. I know that sounds horrible, but often there can be a lot to catch. We've usually found them still in the trap or very nearby (so at least they don't crawl off and die under floorboards and smell).
https://www.amazon.co.uk/Rentokil-PS...rds=mouse+trap
The one time I found one of these pushed out of place with no mouse in it or nearby I suspected a rat, and we put a rat trap down and caught it (the give away was an apple that had been taken from the fruit bowl on the table and was half eaten behind the sofa!
We got one of these for a rat and it also works with mice but obviously you need to not let your hamster out in the room with one of these around. It also kills instantly so is reasonably humane.
Smearing a bit of chocolate spread at the back does the trick.
https://www.amazon.co.uk/Pest-Stop-P...words=rat+trap
It's not cheap but very effective. You need big batteries for it. You just tip them out next day (possible without looking) and re-set it and keep doing it till you stop catching them.
We only had the one rat which had got in from outside and had to block up a hole in the wall. Mice can be quite big families though. At this time of year you often get field mice coming in.
Anyway - I'm afraid we just went the elimination route. I did try humane traps once but it took a while to catch anything - some of them were clever and flipped them open, and they can actually die of fright if trapped in them overnight anyway. Plus it takes a long time to catch them all that way.
Those are the only two effective methods I've found. It's why Charlie is in a cage with 7mm bar spacing. If a mouse did get in the cage it could pass on Tyzer's disease. Don't panic just now though - you could move the hamster cage to the bedroom maybe for now and set traps (depending on where you think the mice are coming out). It's not always easy to see droppings or see signs of them but you can often hear banging noises in walls and scratching noises. Sometimes the only sign we've had is finding something chewed that's on the floor, especially near the kitchen bin.
If it's a rat they do make a lot of noise and banging in the roof or walls and will often steal food a mouse couldn't get in its mouth (eg an apple).