Grain, especially corn is a big offender for many things, most commonly skin issues. In the long run good food ends up being cheaper than vet visits & medication. Good chance you'll see the dog perk up a bit with a better quality food, I had that happen with a 10 year old foster turned family member a few years back. She came to me overweight, with bad teeth, pretty obvious diet of table food when she wouldn't eat the Wellness for the first few days.
By the 3rd day she figured if she didn't eat what was given the younger two guys would continue to chow down on her portion. After about 4 months she lost the excess weight, & was keeping up with a couple dogs way less than half her age. Having a better physical condition for being knocked out for dental work I had that taken care of, after a day of rest it was like a second puppyhood for the old girl.