As Marcel said, your "image" is how people see you, or your reputation. "Keeping up" your image means maintaining it, or keeping it positive. So "I have an image to keep up" means that I have a certain reputation which I need to maintain. People think of me as a particular sort of person, and I want to make sure they continue to think of me that way.

"I have an image to keep up" often means "I am an important person, and I must behave in a dignified way." For example, "It's an outdoor barbecue. Can't you just wear comfortable clothing, like a tee shirt and blue jeans?" "No. I have to wear a dark suit, with a coat and tie. I have an image to keep up." More generally, it means "People expect me to behave in a certain way. I don't want to disappoint them."

Your "image" is how people see you. Your personality, your beliefs, your attitude, your views on topics. Mutt - a dog with parents of different breeds. Mutt - an insult essentially calling someone lesser. If this is from a story, for example, of the Italian mafia, someone of full Italian blood may call someone a "mutt" if they have one Italian parent and one parent of a different heritage.