Charger Port Getting Very Hot

Hey,

I bought a refurbished MacBook for college and I've been having problems ever since. First, the hard drive failed within days of purchasing. Then the left click permanently stuck. Then the charger failed.

So here I am now, with my second charger. And the part of the charger where the light is, plus the port, is getting really hot. More than I would expect it to be. Also sometimes it recognizes that the charger is plugged in, and gives the MacBook power, but doesn't actually charge. Sometimes it does not even recognize that it is connected to the computer. I bought this one used from eBay. I've asked for a refund, so I'm returning it. However I'm now afraid that it's not actually a problem with the charger, but something is wrong with the computer or the battery that is causing the chargers to fail.

PS: I've looked at the prongs inside the charger and I compared it to the old charger and they're different. The middle right prong isn't as long as on the old charger. Could this be my problem?