Calorie in vs Calorie Out

When eating the absolutely most minimal calories (which you are) - I always absolutely recommend to eat at least a portion of them back.

its not that you wont lose weight.its just that its not healthy not to eat them back(you arent getting the nutrients you need) since MFP is set to include your deficit without exercise,so when you work out you create a bigger deficit but that can be unhealthy. I mean there are days you can decide not to eat them back. but whatever MFP gave you as a calorie goal is what you should be netting(after exercise) so if your goal 1300 and burn 300(1300-300=1000 net) you either eat them back,or you can eat 1600(1300+300=1600) and then burn the 300 off.

you have to fuel your body and your workouts.But most people will eat back 50-75% of them back because exercise calories can be overestimated and that will give you some wiggle room. once in awhile its ok to do but not every day,even though Im sure there will be people here who say they never eat them back.

I usually eat extra and then burn them off,for me its easier that way I wont have to worry about eating them back if Im not hungry as well.

It isn't true that if you workout and don't eat enough you won't lose weight. It is, however true that if you workout a significant amount and don't eat enough that you could die. But unless you are working out more than a couple of hours I wouldn't worry about that.