Can i burn fat and gain weight at the same time?

Im 15 and ive been trying to gain weight. I eat a lot of heathy, high calorie, high protein foods and work out mostly my arms and legs consistently. Ive gained a few pounds so far and its going pretty well but my problem comes with fat. Im 5'8" and 115 pounds so you can imagine im very skinny. Also, im not in perfect shape. I have a little bit of fat on my stomach and arms that prevent them from being toned. I have friends that are as skinny but they have 6 packs and veins on their arms and stuff. I feel like if i continue to gain weight there will always be that but of fat. This is why im wondering if i can somehow lose that bit of fat which ive had for years. I love running and plenty of sports but ive limited it to just when i play outside (usually ab an hour 5 times a week) because i dont want to burn too many calories. Since im young and have a fast metabolism its taken me a long time to gain the weight i have in a healthy way and im not sure how i can lose the fat without contradicting it. I know it seems too ideal but im just wondering. Any general advice would be great too as im fairly new to working out. Thanks in advance.