Hello and welcome! As a relatively new rider myself I don't often feel justified in giving advice, but it just so happens that my horse cutting corners was the first real struggle I overcame while beginning riding last year

If a more knowledgeable rider wants to add on or critique my understanding please feel free! You know what they say about the blind leading the blind

So first of all, understanding the concepts of a few components of riding will help immensely! Don't worry if the information doesn't 'click' yet. I know that I personally often have to take some time riding for myself and sometimes even experimenting in order to gain a better understanding of proper mechanics and building that intuitive 'feel'. Just keep things in the back of your mind as you go along and try to apply what makes sense when it makes sense. And soon more and more concepts will begin adding up for a more holistic understanding.
I had to work through 2 main issues before I could ride corners well (and it took several months of weekly lessons for me to really 'get it'). The first was honestly just using enough leg to keep the horse moving forward with enough energy, especially since my leaser is quite prone to laziness. Often, school horses tend to be duller, so making sure you're being firm with your forward energy helps send the message that you're not just a passenger, you're calling the shots on tempo and direction. Your bit about keeping your pony trotting makes me think that you may need to get a feel for adding more leg as needed

From there you have to have correct contact, and this is what will really keep you riding through the corners well as far as navigation. The outside rein is essentially what keeps your horse from "falling in." For example, if you're riding a circle, the tension at which you keep the outside rein basically signals where to "stay" to the horse. While you're taking and releasing on the inside rein to signal the horse to turn, it is the steady outside contact that provides the horse an 'allowance' and 'holds' him in place from following his nose and just making the circle smaller, if that makes sense? So if you apply this to a corner, which should essentially be ridden like a partial circle, you will be signaling to turn with the inside rein, while keeping the horse in place with the outside rein contact. Soft, steady hands are important here. Also, horses are taught to move off of pressure and ideally bend around your leg, so when you apply inside leg in addition to proper contact, a good horse should not move to cut the corner because you're also signaling with the inside leg to not exceed that 'boundary' you've set, as well as help them bend correctly around the curve.
Also important to note is to make sure you are looking in the right direction! Turning your head, even slightly, shifts your ENTIRE balance, and believe me, horses and ponies are extremely perceptive of these minute changes.
Like I said, don't expect to be able to understand everything perfectly from the get go, and even more frustrating, even when you do have a better theoretical understanding, don't feel bummed when it's still challenging to execute. It's all a learning process and things come together in time. Best of luck!