Magsafe 2 connector not connecting & getting hot

I recently bought a genuine replacement 60W MagSafe power adapter (with MagSafe 2 style connector) for my MacBook Pro (Retina, 13-inch, Late 2013) from Apple, after my original power adapter frayed and became a tad unsafe.

I haven't had the new power adapter for long, but I have been starting to have problems.

Firstly, it takes about 5 or so attempts of connecting and disconnecting it before the orange light will illuminate/my Macbook will start charging. I googled it and tried to reset the System Management Controller to no avail.

That is inconvenient, but it's not necessarily the main issue. The big problem is that the charging connector is getting very hot. Too hot to touch (and probably outside of regulations!) after it's been connected for a while. In an attempt to stop this I cleaned the connector and the port, but once again, it didn't stop it.

If you've got any solutions, I'd love to hear them, and thanks in advance!