Can cortisone inj to hip be done along with mri to avoid 2 procedures?

I doubt it: The hip is a hard joint to reach with a needle and doing cortisone injections of the hip is new. It is normally done with ultrasound guidance.

Possibly: The MRI study involves lying still on a table in the radiology department while the machine takes the pictures. Your hip injections would usually be done by your orthopedist as a separate procedure in the office or clinic. You could ask your doctor to schedule the MRI and the injection on the same day, unless your doctor wants to see the result of the injection before ordering the mri.

Maybe: If your Doctor has ordered an MRI with an arthrogram, then cortisone can be administered into the joint with the arthrogram. However, your doctor may have ordered an MRI without contrast, and the cortisone injection may be intended for treatment of the tissues around the hip which are outside the joint. Tissues outside the joint will not respond to cortisone administered into the hip joint.