Would removing the back-board of cupboards help against mould/mold

So you have a couple of options that will minimize the mold right away:

1. Wall boards.
2. Large stone sheets - I have put up a few granite systems and they are not getting any mold.
3. Larger tile.

If you choose wall boards or large stone sheets then you still need to caulk. There are caulks - not usually sold at the big box - that are virtually mold proof. If you want something from a big box then go with GE Silicone II. I have used it in my own house and never any mold.

The key to a moldless shower is that you only let water stand/sit in the pan. So you need to plan on your transitions and things like soap dishes. If you have a nook that is built in the wall and you see mold over it - the problem is the nook not the area directly over it.

Also I have people show me their showers and it is newly caulked... But their tiles are so old and have been scrubbed (probably with the wrong type of sponges) that they have no facing. Meaning water may penetrate just a little each shower. Without enough time to fully dry - mold.

So I get people who want a mold-free shower (and tile) and I always give them the same advice. Big tiles until you hit 4 feet. After that we can go smaller or even mosaic. The less grout the less issues. Then the key is an epoxy grout. A true epoxy grout is basically a plastic after it dries. They look great, they are mold free... So why aren't they used more...

Well the first time I used an epoxy grout was on my backsplash (1 inch mosaic) in my own kitchen many years back. I wanted to test it before I did a shower. Holy cow is the epoxy grouting like 5 times the work as regular grouting. Once you mix your A and B you have about 15 mins to get the grout in and then it slowly hardens to unworkable at around 1 hour. So you must do small batches and have a helper. But man does my kitchen backsplash look good. I have done 5 full showers using epoxy and people are pretty wowed. It is certainly worth it for a shower. You will make up your install time in less than a year with less cleaning.

Note: You should not buy "ceramic" tiles for shower - nor should someone sell you these. They are more porous and will hold water after facing deteriorates - which it will faster than the same quality of porcelain.

Another note: Only have cement back in shower - no drywall of any kind. A Kerdi like system is even better but a well installed HB shower with plastic under it that flow to pan will do.

It's hard to tell from the picture but it looks like it was tile mounted to a wood board. Wood is one of the worst materials for behind the sink. If going to replace with tile, use a concrete backer board instead and consider using tile trim caps to cover the top. Otherwise use a solid stone or stone like material.

And make sure the caulking is in good shape. Every couple of years it needs to be stripped out and replace. If it is starting to look old and cracked, it's been there too long. No mater what the material, water under or behind the splash with always cause problem.s