When my knee was terrible about 8 yrs ago, after all the MRI and Xray business, the "suspected" tumor the Ortho thought I had was not there. So he gave me a Steroid+ Marcaine (like lidocaine) injection into the top of the knee on the femur side.

It did stop all my pain for a while.

The steroid shots work in joints acutely and last a month or 2.
If the problem resolves, that is nice. But if whatever is causing the problem continues, well, then they wear off. My knee shot fixed my severe pain back then. That did not come back. It was some kind synovitis inflamed from the knee strain when it dislocated.

That link I gave you shows where the bursae are in the hip.
It is also possible, if you have had a fall on the hip or have taken prednisone orally in the past, that you have some necrosis of the bone going on. I had a patient who fell on her hip playing tennis, and it started a cascade of hip bone loss and she had to have a hip replacement at 45! People can get this slow necrosis also from steroids. It is a rare but potential risk.
People with asthma, or other allergic things, get those Medrol dospaks, and may later get hip deterioration. Not common but it does happen. If you have low bone density, it can also be a risk in younger patients. (this is called osteopenia).

But most things are bursitis, or strain from running. If it were me? I'd give up the running for a while until you heal up.

I don't think steroid shots work at all for MP. That is a nerve malfunction, or compression, not inflammation.