My 2001 hyundai elantra power steering is hard to drive, I put p.s. fluid in it and it does better steering wise then it goes back to being hard to steer is this the pump or leak? or does it matter.

Check the attached links,similar models to yours power steering belt replace.
"I hope this helped you out, if so let me know by pressing the helpful button. Check out some of my other posts if you need more tips and info."

How To Install Replace Power Steering Belt 2001 06 Hyundai Elantra

Accessory Drive Belts Replacement 2001 Hyundai Accent

How To Install replace Alternator Belt 1999 06 Hyundai Elantra 2 0L

Replace the power steering belt. Make sure that it is tight.

Check the power steering fluid

It sounds like you're probably low on power steering fluid. I think the power steering fluid reservoir on the 2001 Elantra is on the passenger side of the engine compartment near the front. It has a removable cap. On the side of the reservoir there should be two lines. These are the minimum and maximum fill lines. Add fluid to bring the level up to somewhere in between these lines. Make sure you use an approved fluid for the Elantra. Your owners manual will probably list a power steering fluid type or you could go to an auto parts store and they could look up the correct fluid for you. I hope this helps you out.

Genuine Hyundai ATF. DO NOT use any other fluid. I say again DO NOT use any other fluid even if it claims to be compatible. Hyundai says there is no substitute. Drive to your local Hyundai dealer and purchase the fluid there and take no chances of destroying your transmission to save a few dollars.

your power steering pump has a cap on it.you have cold mark and hot mark.when car is cold engine not running.when you remove power steering cap it should be on cold mark.if not add until you get to cold mark on cap stick..dont over fill or you blow the seal.now when engine is hot been running.turn off car check.power steering cap should read full hot.mean level is correct.when you first check power steering fluid and it was real real low.you got a leaking high pressure hose.or power steering pump seal is leaking.if all check out good.your problem could be low tire pressure.worn ball joints.and tie rod ends.rack and pinion steering gear binding.

check the fuse for power steering. It might be electrical.

This is easy to fix. Go to your local autoparts store and tell the year model & motor size. They will give you the proper hose to replace it with, plus ask for the fluid that you will need to pour into the pump tank. Two fitting to take off, & reinstall. Top off fluid level, then crank motor, and turn steering wheel from right to left a few times. Sometimes you might have to do the turning alot to get the air back out of the lines. Later on check fluid again, and top off if needed. I know some autoparts stores will help you in the parking lot?????

I have had this issue but it was resolved by power steering fluid.

If it happens mainly when your car is cold, it is a common symptom known as "morning sickness" or "black rack disease". Check your power steering fluid to see if it has become discoloured. If so, try flushing the system and replace the fluid with an approved power steering fluid. If this does not solve the problem, you may need to replace the rack and pinion steering unit.