Yes, you can get the influenza vaccine and the COVID-19 vaccine or booster at the same time, and actually, I recommend doing exactly that. You might as well get them out of the way!
Getting both vaccines at once does not decrease the efficacy of either and does not cause any changes to either vaccine. However, because you’re getting two vaccines at once, you may feel a little crummy for the next few days. No, that doesn’t mean the vaccine made you sick or that your body is going to have a problem with the vaccine. It just means your body is gaining immunity and using a lot of energy to create antibodies.