One problem related to power steering pump has been reported for the 2013 Hyundai Elantra. The most recently reported issues are listed below. Please also check out the statistics and reliability analysis of the 2013 Hyundai Elantra based on all problems reported for the 2013 Elantra.
My (and my son's) 2013 Hyundai Elantras both have begun exhibiting the failure process that will inevitably result in the loss of the "electronic assist" power steering. Our Hyundais now have a vague dead-spot in the steering when the steering wheel is moved off center at any speed. You can feel it the most when the vehicle is parked. The "dead-spot angle is getting wider and wider as the rubber slip-joint part inevitably disintegrates this rubber bushing is a shear device between the mechanical steering shaft and an electric motor that powers the Hyundai's steering rather than traditional hydraulic pump power steering found on most cars. My neighbor's Hyundai went fully through the failure process that resulted in no power assisted steering and a very expensive repair. The failure of this $1. 25 (retail) part is so common in many Hyundai models like the Elantra, that the parts department keeps a box of them constantly on hand for the inevitable failure of this part in every Hyundai model that uses it. It's a huge profit producer for their service department, so I can understand their reluctance to redesign this part. However, I can't currently afford the hundreds of dollars of labor charge that Hyundai demands for the replacement of this defectively designed part, so I don't let my wife drive the car because she might not be able to react to the loss of steering assist at high speeds. I drive the car as little as possible. Hyundai is well aware of this defect and had a recall for the same problem on 2010 models but ended the recall when they probably told the NHTSA that they corrected the part. In actual fact, they didn't fix the inherent defect and if you do a search online you will see that many people are still being endangered by this defect.