Apologies in advance for the lengthy post - I’m a cleaner, so have spent a long time trying and testing various mops!
I agree with @Porridgeislife - e-cloth mop system. A lot of people think they just move dirt around, but that’s absolutely not the case IF you use them properly. Of all my clients, there is only one who I use a traditional mop (plus long handled deck brush for) and that’s because they have riven slate floors which are too textured for a flat mop to be effective.
Obviously you need to vacuum properly first (or sweep, but vacuuming is better), same as if you were using any other type of mop.
My current e-cloth mop was bought in May 2021, and is still in perfect working order - telescopic handle fine, Velcro firmly attached - and it is used a LOT! It came with one pad, and currently costs £18.29 from Amazon (free delivery if you have Prime). Branded spare pads are £££, and I had to look elsewhere because I get through so many a day. The best I’ve found are in Home Bargains - Pro Action brand, 2 for £2.49. They fit the e cloth mop head perfectly, and don’t shrink if you wash them at 30 degrees or less.
You will need a spray bottle containing your preferred floor cleaning product - simply spray the floor, then start mopping. You do not need to saturate the floor with spray - less is more, think grip not slip. It will take a while for you to work out how often you need to change the pads as you clean, and it depends on how often you actually mop of course. As an example, one of my clients has a 30ft x 13ft white stone tiled kitchen/diner - if I mop every week it’s 2/3 pads, if fortnightly 3/4.
The great thing is you can just do the most important bits (like under worktops/by entrances) as often as you like without fannying about filling a mop bucket and emptying it. Uses lots less water, and reduces product cost.
Yes, you will have to buy the mop itself plus spare pads - but the pads are washable, and last for a really long time.
As a bonus, you can flip it up to clean skirting boards/kitchen baseboards, plus use it on wall tiles and big windows. I’ve also used it on heavily greased walls (with sugar soap), and to remove cobwebs from ceilings/walls (dry).
I’m suggesting the e-cloth mop because mine has lasted for so long - there are cheaper alternatives I’m sure. But I wouldn’t advise any mop with a built in spray bottle - happy to say why if anyone asks.