I am in the military and do not live in my house. I rent it out for $1560 per month and pay a mortgage of $2126 a month. Can I claim the extra $566

No, it doesn't work that way.  You will report the annual income you receive in rents.  Then you will deduct your property annual deductions: mortgage interest, insurance, utilities, repairs and maintenance, property taxes, etc. The total amount of your mortgage payment includes principle and interest, and maybe property taxes and insurance.  The principle part of your mortgage payment is not deductible for income tax purposes.