I have tried using wood cleaner and ring away.
But honestly the stains (I think by water) have got worse.
How can I get rid of them?
Woodworking Stack Exchange is a question and answer site for professional and amateur woodworkers. It only takes a minute to sign up.
Sign up to join this communityI have tried using wood cleaner and ring away.
Neither one of these was an appropriate product to use as there's finish actually missing.
How can I get rid of them?
'Spot refinishing' is what you're looking to do and this is notorious for being tricky to impossible for the amateur to pull off. It's not at all difficult to do, the hard part is in getting it not to show. Even pros can struggle to do this, just to put it into perspective.
Plus, in the UK some of the product that would be most likely to work for this aren't sold or are only available through specialist suppliers. This limits your options or makes those available more expensive than you'd probably like.
If the piece has value to you, and you want it to look its best
I would recommend you give strong consideration to stripping the whole top and refinishing it. It's a lot more work but not really difficult work, and much more likely to give you a result you'll find acceptable. Plus, a top where all the finish is of equal age will do better in the long term.
Please see note below.
If the piece is not that important
You could just leave it as it is. The small patches of exposed wood are not of any great significance except aesthetically. The table won't self-destruct, even if exposed to modest amounts of water from cleaning, spills or sweaty glasses.
If you find you just can't live with them I would recommend applying a few coats of wiping varnish. Buy a small tin of the cheapest varnish you can find (a pound shop should have something), some white spirit (again, pound shop if you only want a small amount) and combine the two in a clean jar. More on wiping varnish in this Answer. You'll also need some fine abrasive paper (320 grit is probably about right) and/or steel wool (000 or 0000 grade). You can use kitchen paper as your sole application tool, although scraps of lint-free cotton cloth are preferable (careful when disposing of either1). Although dilute varnish is fairly innocuous you may wish to wear rubber, vinyl or nitrile gloves as well so you don't get any on your fingers.
A note on removing old finish from furniture
If you choose the first option I strongly recommend you do not remove the old finish by sanding. As I've covered in numerous previous Answers, sanding is the worst way to remove old finish2 and should be reserved for situations where it is the only available option – it should be be considered the method of last resort, i.e. all other options are preferable.
Chemical strippers should be thought of as the no. 1 way to remove old finish. While stripping is undeniably messy and takes some time3 it is preferable for a few reasons in almost all cases. Every properly trained pro will strip old finish by preference, if it's necessary to remove it entirely and they can't just refresh what's there.
1 There are a few ways of doing this but the simplest is probably to dry flat until they've gone stiff, then they're safe to throw in the rubbish.
2 In addition to the risk of damage to the furniture itself — including but not limited to sanding through veneer, rounding edges and corners (both easily done even with experience and while being careful) — it converts the dried finish, plus any dirt or contaminants in/on it, into a fine dust which is a health hazard.
3 Most modern consumer-level strippers work more slowly than the strippers available previously (actually up until just recently) which relied on a very effective solvent which is now restricted in most markets.
The thing with finishing is for it to look good you really want it in one coat and without knowing what finish is already applied I would recommend refinishing the top completely. You could just patch it with some new finish but it will stick out like a sore thumb.
I would sand back the whole top starting at 60/80 grit and making you way up to 320 grit. Wipe over with white spirit allow to fully dry. Check for any grain rise (run you hand over it, if it feels bumpy light sand again with 320 grit). Then select your finish.
I would personally use a Danish oil as it brings the grain out and has a protective quality when it fully cures. However, there is always varnish and other types of oils. I would stay away from wax if you plan on putting anything with heat or moisture on it.
Sanding will also give you the additional benefit of removing all the other scratches from the surface giving you a flawless finish.
While using chemical strippers is easier, there are some fundamental things you should know about it before you go down that route.
Without proper protection you can run the risk of serious respiratory system damage as well as various other health issues. If you choose to use a chemical stripper make sure you use something that doesn't contain Methylene Chloride in it. Even products that have not got Methylene Chloride present in them can still be damaging to the environment.
A dusk mask can keep you safe from the dust produced when sanding. It will not protect you from the vapours given off by the stripper; this requires an M class respirator.
Either method would work for what you have, but you should be aware of the risks involved to the object or yourself before choosing either way.