I spent over 10 years maintaining pools (up to 30 at a time) and I've repaired hundreds more.
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Firstly, I would NEVER advise leaving the pool empty for any length of time. As Jim says, the water helps to keep it stable and stops the grout and tile adhesive from failing. Don't forget, an average 8 x 4 pool contains 40-odd tons of water, which helps to keep the sides apart!. Also, if you empty the pool, and it rains, you will have a big puddle of dirty, often sandy, water in the bottom. This will be a magnet for any rubbish, leaves, plastic bags etc that are blowing about. The resulting filthy cocktail will stain the grout, whilst it's busy being a breeding ground for mosquitoes. The effort of cleaning all this out alone will make you wish you'd kept it full, trust me! When the pool is full again, you'll very likely need to replace the pool pump, as it will have seized up through lack of use in a damp environment.
So, as you can tell, I favour keeping pools full and topped-up with water and the necessary chemicals. You really need someone to keep an eye on it at least once a fortnight in winter and once a week in summer. All the private pools that I used to look after were visited once a week from the end of September round to April, and twice a week during the summer. The pump running time can be reduced in the winter, but as already mentioned, the pump ideally needs to run daily, if only for a short period. I run my pump/filter for 2 or 3 hours in the winter and 7 or 8 hours in the summer (depending upon bather load). As Jim says, it's not essential to remove the dirt, but a backwash every month or so is a good idea, even if only for 30 seconds or so, because if the filter gets too clogged up it will put extra pressure on the pump and the sand filter, and could cause a leak.
I am not a lover of pool covers - to me the water never seems to be so "happy" if it can't "breathe" and so (purely my experience) the water balance is more difficult to maintain in a covered pool. That said, obviously I accept that a cover keeps a lot of debris out of the pool. Another problem with a cover is that rainwater often gathers in a big puddle in the middle on top of it, putting an enormous strain on the cover's fixings, and creating a filthy cocktail like the one mentioned earlier, which is difficult to get rid of (if it's big enough, you can't lift the cover to tip the water away either).
Summing up, I think overall it will be beneficial for you to keep the pool going year-round, and to find someone to keep an eye on it (maybe a neighbour?). Apart from anything else, if you arrive to an empty pool, by the time you've cleaned it, filled it, added various chemicals to balance the water and waited for them to work, and waited for the water to warm up, you'll have missed a significant period of your holiday when you could have been enjoying the pool. How much nicer to be swimming a couple of hours after your plane lands!
Please feel free to send me a pm if you've got any more questions.
Geoff.