There are two grounding rods outside my house that stick out above the ground. Besides being a pain to mow around, the longer (bent) one is quite easy to trip over. Is there anything I can do to get these completely buried?
Never assume a rod that is not being is not needed!
If the #6 copper has been ripped out it probably needs to be replaced. The short one is easy dig around it and drive it below the sod. Next dig a trench to the other one and I would use a piece of metal pipe to straighten it out and drive it below the sod. Using the acorn nut on the rod connect #6 or larger copper wire rod to rod to your service panel. I have worked on a few homes that for some reason the ground was removed (this can be hazardous to your health)
So unless you can verify you do have a solid ground drive them a bit further. If new rods were driven they can be pulled up or cut off.
Before you cut them or do anything else to them, I would dig around them and see if there are clamps on them with a copper wire. From the picture, I see you have you grounding wire going in to the ground and most likely connected to those rods. Dose are very important for your and your family's safety.
I would suggest that you use a shovel to dig out a pit maybe 6 inches deep surrounding each rod. Examine for an attached wire, and if you find one then dig deeper, so that when you pound the rod into the ground you will not cause the wire to slip loose.
Pound each rod in until the top is maybe 2" below ground level, then fill in the hole.