Should I eat the calories added from exercise

I've eaten back all of my carefully-estimated exercise calories all through about a year of loss from class 1 obese to healthy weight, and for going on 8 years of successful maintenance at a healthy weight since.

That would be a reasonable choice if someone has followed MFP's instructions and set "activity level" in their profile based on daily life (job, home chores, etc.) excluding intentional exercise.

Syncing a fitness tracker to MFP somewhat compensates for inappropriate activity level setting, as long as a person enables negative adjustments in MFP.

If someone tells MFP they want a very slow weight loss rate (like half a pound a week), and doesn't do much exercise, it's fine to let that small amount of exercise make the calorie deficit bigger, to achieve faster loss.

Someone who sets a very aggressive loss rate target (more than 0.5% to an absolute max of 1% of current weight per week) and does a bunch of exercise is increasing their health risks, plus increasing the odds of failure to stick with it long enough to lose a meaningful total amount of weight, not to mention learn new habits that will let them keep the weight off long term.

In between those extremes, it's a judgement call about how much health risk (and failure risk) the person likes to take on.

Personally, I value my health, and I did (do) want to accomplish my goals. I averaged about a pound a week loss when I was losing. It was a bit faster than that average at first when I had more available to lose, and intentionally very slow when I got close to goal weight.

That worked great for me. I was overweight/obese for around 30 years, and have now been at a healthy weight, as I said, for around 8 years. My quality of life has improved a lot as a result.

It's also possible to get a calorie goal from an outside calculator like this one, that averages in your exercise plans:

https://www.sailrabbit.com/bmr/

One way or another, exercise burns calories, and one needs to consider how to manage that. Any of these methods can work IMO, with conscious attention to reasonable goals.

No matter the way of estimating calorie needs, it's very important to stick with a plan for 4-6 weeks (or at least 1 to 2 full menstrual cycles for those who have them) in order to see a reasonable actual average loss rate. Weight loss isn't linear, so we need a few weeks to average, in order to know whether a plan is working.

The estimates will be close for most people, but can be high or low, in rare cases surprisingly much so. The estimate isn't Pure Universal Truth, it's just a statistical average calorie need for superficially similar people. Individuals can vary from average.

Best wishes for success - IME the effort is worth the reward!

Miserable is not sustainable.

If you’re hungry and/or feeling exhausted?

Eat those calories.