Daily Calorie Goal changing with your workouts...

Is it necessary to eat back the calories you have earned back or is it better to leave a deficit?

If you are using MFP to calculate your calories, it has already built in a deficit and assumes you will eat extra cals when you exercise.

Www.shouldieatmyexercisecalories.com

If you're burning that much, you should at least eat half of it back. its not safe to eat less than 1200 calories a day- and if you think about it like this, by exercising the equivalent of burning 2400 calories, you're really -1200 calories for the day if you only eat 1200 calories.
make sure you're staying safe- if you eat less than 1200 calories a day, your body can enter starvation mode and pack on the pounds from anything you eat- I'm sure that's the last thing you want!

Just remember that mfp calories burned estimations are exaggerated by about 25% for most. I would eat back about 1/2. Your body is not going to enter starvation mode if you eat less than 1200 calories, unless you have like 6% body fat and you do it all the time. Starvation mode is a complete myth. You are not going to gain weight either. When I was eating 1200 or less consistently I lost some weight in the beginning and then as my workouts got more intense it stalled. I've upped my calories 100 at a time and I consistently lose 2 lbs/week. I burn about 600 calories per workout/4-5 days a week. I stick to my cal goal even on workout days but if I go over 100 or 200 I don't stress it. I just upped my goal again from 1350 to 1450 because I added an extra day to my personal training every week from 2-3. There is no way that I would eat 2400 calories or even above about 1900.

You probably should eat them back, but make sure you are really burning that many calories. If you have a way of measuring your heart rate during the workout, there are plenty of sites that you can enter heart rate, age, weight, and height and it will estimate calorie burn. Or you could get a heart rate monitor.

Some exercises are pretty accurate for me on MFP - elliptical is spot-on to within 5% or so, verified by heart-rate calculation, the calculation on the machine itself, and MFP's calculation. But I set the resistance on the elliptical to high and run like I'm being chased by rabid wolverines to get my heart rate up to the max cardio range, then focus on keeping it there. If you leave it on low resistance and go slow, MFP's numbers are going to be ridiculously high.

Others are really high to start with.

I'd experiment with eating back, say, half of them. If you lose weight at your goal pace, that's what you need. If you aren't losing weight and your exercise performance continues to improve (faster, higher, longer, stronger) you need to cut back calories. If you aren't losing weight and your exercise performance is decreasing (harder and harder to maintain the same effort) then you aren't fueling your body and it can't support good workouts - eat more.