Why is my iPad 4 charging in a cycle?

So, for the past 1.5 weeks, I've been letting my battery drop down low up to 10% and often less, because I had no charging spots on the way from school, to courses, and home. I only had them at school and at home, but when my battery got to the point of 10% or less, it usually didn't really have that much of an effect towards the device itself.

Then, for about a 3-4 days now, it started turning off when it's on 10%, sometimes on 8%, or on 6%. It's pretty common on iPhones (mine's an iPhone 4). I was starting to get worried at that point, because when I re-charged it, the stage when it pops an Apple logo on the screen becomes way longer than usual. It could take >8 minutes, which gets me pretty apprehensive.

And just last night when it went off again on 6%, I happened to be bringing a power bank. So I charged it with it and it worked, because it got to the point when it showed on my lock screen, but my power bank suddenly went off. Of course the power got cut off and had to turn it on again. But then when I did, it didn't work like the first. I thought maybe the electrical current (Volt) wasn't powerful enough to charge an iPad because it's usually used for me to charge my iPhone, but still...it's weird. When it worked in the first, why couldn't it during the second?

When I got home, I quickly charged it in my room's charging socket, but it just worked into this really weird charging cycle.

First, it showed on the "battery picture", then the "Apple logo", but after that it turned back to the "loading logo" (as if it's losing in the battery power and turning off, but it's not, because I'm charging).

Secondly, I started gaining hope because it's been going on in that cycle for hours, when it started to stay in the "Apple logo" for some time, but then it went back to the "battery picture". So I thought about the tiny, little slit I had in between the end of the cable towards the connecter, to the connecter itself. I had it wrapped up in tape for quite some time now (which is more than a few months), so I decided to open it up and check on it. The slit got bigger, and I put two and two together.

The connecter started to get heated every time I charge for a long time for the past week, especially the latest 3-4 days.

So the real question is: is there any chance of it having a fault in the system (because of it turning off at 6-10%), or is it just the cable that's building up this "charging cycle"?