This is okay, we all start from somewhere. I recommend you take a picture of your normal push-up form if that is also what you are struggling on. Then compare that to pictures found online of the perfect push-up and try to adjust to the perfect form by contacting your glutes. This will not fully fix your core strength, it will just expose you to perfect form so you don't face issues in the future. If you find your core is very limited, you can do planks regularly (2 times a day to start off with) but do not go too far in each set because it can be very demotivating, just stick with a duration you believe you can stick to consistently and your core strength will improve!
