Should I eat the extra calories I gain through exercise?

Is it cardio? If so, nah you don't have to if you don't want to.

Edit: Even though you're impatient (as you say) Don't overdo it. Take your time, if it feels like you're starving it's easier to revert back to old habits. If you're satisfied and don't mind, you don't have to eat it all back.

Assuming you are using the MFP method, yes you should eat back your exercise calories. This is not to promote weight loss but to promote health and to ensure that most of the weight lost is fat. When you burn calories, you burn from both muscle and fat cells. This is always the case, but the ratio varies based on a lot of different factors. The size of your calories deficit is a large one. MFP already creates a deficit for you, so if you eat all of the calories MFP recommends, you are still eating less than you need for maintenance. When you exercise, your required calories to maintain increases, so if you don't eat an equivalent amount, your deficit will grow too large. This can lead to lower energy and can also cause your body to try to hold on to fat reserves to offset a perceived food shortage (if the large deficit is maintained for long enough), such that you begin to burn muscle instead. You will still lose weight this way, but much of the weight lost will be muscle, which you want to keep. 1200 calories is already pretty low for most people, so dropping much below that is not recommended.

Eat. You need at least 1200 cal a day to avoid "starvation mode" so if you burn 300 cal from exercise you need to eat at least 1500 cal. You'll soon find what works for you. Good luck x

That's false though.

I was most successful losing weight when I ate back most or all of my exercise calories and maintaining a consistent calorie deficit each day.