Hello
That is quite a bit of weight loss. It's fairly normal for an older hamster like this (nearly 2) to lose a bit of weight, maybe the odd few grams every few weeks or something. One common thing causing weight loss when they get older is toothache. ie too painful to eat hard food, so they're not eating properly. Or another tooth issue (missing or broken or overgrown front teeth eg). Maybe you can check his teeth. Yawning is a good way to have a good look, if he does that now and then? If not, if he's not too wriggly you could gently pull his lips/cheeks up at the sides to look.
If seeing a vet anyway the vet could check. That would show any obvious front teeth problems, but sometimes it's just toothache in the back teeth and nothing to see. Or even an arthritic jaw.
Hamsters at this age often have some arthritis or toothache and do very well on meloxicam (anti inflammatory and pain relief). I would ask the vet about some for him when you go.
In terms of food at this age, supplementing with soft food and other things can help them regain a bit of weight and keep their weight on. So for example you'd still put the usual hamster mix out (they like to hoard and pouch it out of habit even if they're not eating it really) and in addition, put out a few science selective pellets, which are then soft and mushy and contain everything needed. In addition, a pinch of shelled hemp seeds daily helps with extra calories and contains most vitamins and minerals - plus hamsters seem to love them and lick them up. Also a bit of baby food and or porridge (only certain types of baby food - we have info on here about that separately), eg on a teaspoon.
If you get prescribed meloxicam, a good way to give it is drop onto a large pea sized amount of baby food. If it's more than that they may not eat it all in one go but they usually do when it's such a small amount. And it saves bothering them with trying to give it from the syringe. One of mine is on it at the moment and he eats his baby food and meloxicam straight away and licks the plate clean
Another possible thing that might be going on is diabetes. You can check for that even before you see the vet. The majority of winter whites are actually hybrid russian dwarf hamsters (ie genetically a hybrid of winter white and campbells even if they look like a winter white mostly). Ear and eye size and shape and various markings can give an idea that they are part of each species. The two species have been interbred for decades so unless he came from a specialist pedigree breeder he is almost certainly a hybrid.
Hybrid and Campbells are both predisposed to diabetes. Some will get it, some won't. But you can just some ketodiastix from a pharmacy (human ones) and dip it in some urine which will show whether there is any sugar (and/or possibly protein) in the urine. If you can get the hamster to pee in something like a plastic container. Some people pop them in a plastic pet carrier for a while without bedding to achieve that. If he does have it there isn't much you can do except be careful with diet, but it helps know what's going on and help with diet choices.
Sometimes you can guess it's diabetes just from the fact that their urine may be sweet smelling.
So yes I think a vet check would be good - so he gets a look over, in case there is anything else going on, but it may just be old age symptoms like diabetes and tooth issues.
If you can find an exotic vet that is much better as they know more about hamsters.
Other than the weight loss is he still hamstering? Using his wheel, foraging, a little bit active? Seeming ok in himself?
