Firstly, you have a lot of pressure on yourself as a single mum, and a grieving one at that, and all the extra things you will worry about for your children and their psychology.
The main thing that will help you is to be consistent.
Thr next thing that may help you is to change their diet slightly. Cut sugar (this can give them great highs and manic times, and then crashes). Sugar really is like a drug for kids and should be limited. Trust me, you'll see a difference. Up their intake of veg and especially red veg or green veg. Too many carbs can also turn into sugar and can affect their behaviour. Carbs are good, and filling, but just not too much.
Getting up in the morning and making eggs is a normal thing for an 11 year old... they are finding their feet as they are growing into a teen (and although they are allergic - it was just a normal trial that kids at that age do), they love to move on to the next stage of development and experimenting. You can give them an alternative, tell them "in the morning, you can help yourself to such and such and cook it for thr both of you."
You are tired. You're brain is overworked, kids are very astute at picking up on overworked parents or stressed parents, and will be reacting to this. As well as the reward thing, I think you should find a day in the week where someone else takes the kids for a good few hours and let's you go and have a coffee with a friend and just be you, vent, and be a human again.