I want a Cheaper Car

It depends on whether you want a replacement car, or you simply want the most cost effective option.

The cheapest option would be to see if the car passes its MOT in four month's time. If it does, or only needs minimal work, then fine, keep it, otherwise scrap it and buy something that you like. This is likely to be cheaper than upgrading as you have to factor in depreciation and possibly a car loan.

Petrol costs are only a small proportion of a car's total running costs.

The argument of Tax and MOT is irrelevent as either way you will pay this as an annual cost whatever the car you have. The only way you wouldnt need to pay MOT was if your car was a virtually new car. Then the £50 saving a year would be more than offset by the increased depreciation. A better solution would be to look at your driving style (how efficient) and how much you use the car. If possible now that the summers here get the bike out and do some cycling to work. If you live within 10 miles of work its the perfect time of year to start cycling.

I think you need to measure the cost of your motoring in the right way rather than thinking, I have to spend £240 to keep a £750 valued car running for another year.
You'll have to spend that to keep any car on the road and likely more.

If you are spending £30 a week / £120 a month / £1440 a year on fuel there is actually very little scope for cost reduction here, your not doing all that many miles.
All the time this old car doesn't throw any repair bills at you the best bet is to keep it going. Problem is cars are always expensive, having an older one doesn't make it cheaper by as much as you might hope.
Newer car = certainty of depreciation expenses, older car risk of repair expenses. The trick is to switch to a newer car before repair expenses rise too much.