What am I doing wrong?

Since you are working out, you should be eating more.

If you're hungry, eat. Your body is really smart. It gives you little clues to let you know it needs something. If you get twinges that you think is hunger... drink a glass of water... if you still feel hungry... eat a healthy snack. Are you hitting your macros? It sounds like your protein is kind of low. I try to get a higher deficit during the week to give myself room to eat a little more on the weekends "work hard to play hard" kind of thing. No plan is the same for every person... you have to fuel your body for the work you're giving it to do and do what works for you.

Agreed.. Working out = eating more. You may not have enough fuel to get through the day. Leafy greens are a must like Spinach and Kale! Also avocado is a great source of healthy fat! It looks like you may need a larger amount of protein and a little less carbs. Too much sugar leads to quick rise and fall of energy and leaves you feeling sluggish, hungry, and irritable!

If you dont have any medical issues,then for sure you dont eat enough
You maybe be around 1200 calories a day, but with working out like you do, you need more food.

that's why you are tired and burned out. Your body needs full when the motor is running faster/harder.
1200 calories is not a lot and will do when you are sedentary and dont do a lot at all.

I was on 1200 calories for 4 months ( now 1300) and i was not tired, or burned out at all.
I train 6 days a week
Eat 1/4 back of my burned calories
And lost over 5 months almost 75 pounds

And in those months i just had some days i was tired ( with an explainable reason like moving stuff and other unusual activities)

I feel energetic, happy and driven to do things.
But i eat a lot more food than you do on a daily base for 1200 calories.

So my opinion ( if you dont have any medical issues) is that you have to eat nutritious food ( a lot more vegetables and fruit) and eat back also at least 1/4 to 1/2 the calories you burned.

Eat more, for sure.

I started at 178.5, 5' 5.5" and I turned 40 last year. Lost ~53 pounds in 9.5 months. The first 6 months, eating 1400-1600. Then the last few months eating 1600-1800.

Being tired, and low on energy in general is your body's way of saying something is not right. Eat the foods you want - but aim for a higher calorie count. Have some veggies & a side with your 6pm dinner. Increase the size of your snacks a little. Though you indicate the issues happen later in the day, so increasing the afternoon snack and dinner may 'fix' the problem.

Hi, what I´m seeing here is a lower intake of carbs that what you should. don´t be afraid of good, whole carbs, they are your best ally when it comes to feeling satisfied throughout the day. I have lost 55 pounds and kept them off for 2 years already and I have never given up on carbs (the good ones). from your daily diet I can say:
1. your first meal of the day is full of fruits which have good type of sugars but they give you just a brief pump of energy, what you need to add there is at least a slice of whole wheat break or multigrain tortilla, you need this because you will need that energy in about an hour for your workout, also instead of 3 egg whites, add at least one whole egg so it gives you a good fat to start the day off.
2. your post workout meal, make sure the whey protein you are using has a balance amount of carbs too for muscle recovery, not only protein. there are some with no carbs at all and those won't be that helpful after a workout.
3. your lunch looks perfect to me
4. Greek yogurt, then again good protein but here you need to either a little "good fat" source or a good carb source. probably 1/8 cup of walnuts, almonds. in this snack I have one scoop of any isolated whey plus 1/4 cup of old fashion oatmeal. trust me this helps me to continue the rest of the afternoon not feeling hungry and I HAVE KEPT THE WEIGHT OFF and lost fast and gain muscle.
5. just grilled chicken at night won't be enough. at least if you don't want to add more calories try to add a good amount of vegetables, the fiber in the veggies will help you feel satisfied, try with probably a cup of steamed broccoli, asparagus or any other of your preference. I eat them raw or just steamed for literally a couple of seconds so they won't lose all the vitamins and fiber.

sometimes when I go to check people's diary I notice high intakes of sugar, don't focus so much on the calories but the quality of the calories. there might be low calories products but the few calories they have probably come from hidden suggar. a lot of non fat yogurt options in the market may even have as many as 25 grams of suggar per serving so watch out for sugar intake.
remember: don't be afraid of good carbs and don´t be afraid of good fats. it is all a matter of balance and if you are following a good excersice plan you will need them both for good recovery.

Congrats on the weight loss! Id say up the protein intake for sure. Egg whites aren't doing anything for you as the majority of protein and nutrients from eggs are in the yolk. I eat a piece of multigrain toast with a tbsp and a half of natural peanut butter for breakfast and it keeps me going till lunch. Then for lunch, a smoothie with spinach, 1 small activia yogurt, tbsp flax seed, 1/2 cup trop50, 1/2 cup frozen strawberries and 1/2 cup frozen berries blended and a hard boiled egg (around 380 Cals for both). By the time I get to supper I have a lot more wiggle room to eat what I want and stay within 1200-1300 Cals a day. And I don't feel hungry or irritable and I care for my 2 year old during the day and work at night. Definitely play around with more nutrient dense foods! Good luck!