Hi, I'm a recently-retired engineering geologist, with 40 years experience in assessing groundwater and ground issues as they relate to structures (amongst other things relating to ground and structures)
Firstly, it could be groundwater, and if it is, don't try to lower it by much, otherwise you'll dry the clay and it may shrink a bit. That's not good news for houses founded on clay.
If you were measuring the true gw surface in a clay, it would not be going up and down noticeably in a few days - it would probably take weeks because of the low permeability of the clay. Indeed, it would be unusual to see much gw level variation in a clay - sands and gravels, maybe. Clay, a bit unlikely, unless there's some significant gw removal going on locally - and you'd know about that.
Much more likely is water accumulation on top of the clay as a result of flow over the top of the clay, so maybe runoff from the garden, or next door through the topsoil around your house. Given that you see relatively rapid rises and falls of water in your holes, I'd strongly suspect flow across the top of the clay, and similarly drainage away across the top of the clay.
I'd have a look at why this has become a problem. What's changed to give you a problem? If you can identify that, maybe that provides a solution. Has next door built a patio so that rainwater now runs off onto your property? Or do they drain their patio to a drain that's become blocked? Or a garage roof to a drain that no longer works, so the runoff runs across the ground surface (usually unseen through the topsoil) onto your property?. If that is the cause, it's up to them to fix it - there's well-known legal precedence here - Rylands v Fletcher).
https://en.wikipedia.org/wiki/Rylands_v_FletcherFailing that, arrange some drainage to the top of the clay so that the water can find it's way to your garden and a surface water drain. You can do a lot with some gravel, wrapped in a geotextile - the fabric sold in garden centres as weed suppressant (looks like a rough woven textile - usually black). If you can dig a shallow trench in the clay where water accumulates, and lead that trench away to a low spot outside the house footprint Line the trench with the textile, and fill it with gravel. Wrap the textile over the top and cover with gravel. The textile lets water through from the clay to the gravel, but stops fines going with the water flow and evetually clogging the gravel.
I appreciate that the trench solution may be difficult to execute, but either stopping the water getting under your house in the first place, or controlling the level under your house is the way to go.
If you can identify water flowing over the top of the clay towards your house, you can use the trench as a cutoff, along the ouside perimeter of your house, to intercept the flow and lead it away to a suitable point.
The identification of potential gw problems in construction, and their solution, comprised a major part of my working life, in those parts of the world where it rains. Less of a problem in the arid world. Whatever, don't try to change the gw conditions by much, otherwise you'll end up with other problems relating to changing the balance of nature.
HTH, but maybe PM me if you need more.
Andy