Talk to your mom about why you are wanting to eat healthy . make her a list of foods you would be interested in eating, making sure the food you are choosing is cost effective such as carrots. tell her your concerns and go to the store with her to help make healthier choices . eating healthy doesn't always mean having to spend more money.
A lot of times when you eat healthier you spend less. You eat less and crave less so tho the food costs more you spend less overall. My husband and I recently changed our eating and have actually been saving $200+ a month. Maybe do some research of cost of foods recipes etc and present them with that. It's a lot of work but worth it. One site I suggest is weelicious it is healthy food any age can enjoy :-D
i have family members that are in the same situation as you and my heart aches for you. it is not fair. you could try and make your own healthy food out of what you have there. for example..... use only egg whites and make an omelet, do not eat breads, etc. write down what you want to eat. try whole wheat pasta instead of regular. your mom may go for that. but, just because there is bad food in the house doesn't mean you have to eat it. i am sure there is a website that has some healthy meal ideas for you.
when school is back in session try talking to your P.E. teacher or a counselor and maybe they could talk to your parents and give them advice.
forgot the most important part.
i do not know how old you are, but realize that your weight is not what is most important. your health is. you are on the right track by wanting to eat right, but also make sure you are an active child. my kids just wanted to play video games, computer, watch tv, etc. i had to push them outside to play to make sure they got some exercise.
you are growing so you need calories. you will harm your body by going on adult diets. they are not meant for kids whose brains and organs and bones are still growing.