You need to work in some give-and-take.
Perhaps your mother wants to show off a little - either about the car itself, or more likely, about you having that car (and driving her around).
If it's about the novelty of you having a car and driving her around, this will likely pass. This might be a small luxury she indulges in just to enjoy the feel of moving through traffic without being the one at the wheel. Give it a few more weeks and see how it goes. After that, it might be that she likes the functionality or ride of your car - perhaps it has a big boot for groceries. If she is covered by your insurance and can drive, try this:
Here are the keys, Mum. I don't need the car until 5pm. Would you mind filling it up for me on the way back?
100 miles per month works out to about 3 to 4 miles per day on average. If we take petrol at 150p/L and 6 miles/L, that 100 miles works out to about £25 per month if I got my sums right. Topping up the tank a couple of times each month should just about cover that. If the tank is nearly empty, a single tank would likely cover a couple of months' worth of driving.
Meanwhile, it might be nice to indulge your mother from time to time. If you initiate some dialogue as you drive, the conversation might default less to complaints about your driving.
As AndreiROM's comment notes, this addresses "having her impose on the OP's time, the bickering about his driving, as well as maybe getting some money back on the gas used".
If this doesn't go far enough, or if you need to give up your car often enough that the car becomes a communal vehicle - and if you want to address it at a more fundamental issue, try this:
Mum, can we have a discussion about the car? ...
I'm quite happy for you to use the car when you need to, but it was rather a large purchase and I'm losing the sense of ownership of the car while still paying for insurance and fuel and so on - it's starting to feel like a communal vehicle but no one else is chipping in. I'm also struggling to make decent headway on my savings. Would it be okay if we went easy on the car for a bit?
This brings up the topic in a respectful manner while putting the spotlight on the financial aspects.
On the topic of "driving her to pointless places", consider something like this:
I don't mind driving you to places when you need a lift. It's the round trips to nowhere that puzzle me. Instead of all these 10 minute drives everyday, why don't we drive to the park once a month for a family picnic?
The intention here is to change the negative issue of "driving to pointless places" to something that (hopefully) your mother and you - and perhaps others in the family - can look forward to, while reducing total unnecessary mileage.