Something is wrong with my laptop's battery. If it is under warranty (until next month), would they charge me?

I currently have a Lenovo yoga 2 pro. The first issue I had with the laptop was that whenever it would update, it would take HOURS to do so. The last time it updated, it was stuck on the screen, so I had to completely reset my laptop. Then everything is working.
Things were working fine for two months, until one day I realized that the laptop wasn't charging even though it was plugged in. So I went out and bought another AC adapter (which is EXPENSIVE), and it started working again for about 2 weeks. Recently it hasn't been working, so now I have to PUSH the AC adapter into my laptop and hold it there in order for it to charge… But now today, it's not charging at all.
I figured I might as well get it completely fixed since it's still under warranty. But where do I go? And would I have to pay?

If it is determined that the faulty charger socket is due to abnormal wear and tear then this would not be covered by the warranty.
You would need to contact Levono support, there will be a number to contact in your documentation for warrnty repairs

Well so you understand what a warranty is.

Almost every warranty provided from the manufacturer of the product (except very very few), it ONLY covers manufacturer defects. Meaning if the root of the problem was something due to the manufacturing process, it will be replaced for free.

If its found that it was something you (or someone else around you) which was the cause of the damage, it will not be fixed for free. (like if it was plugged in and you tripped on the cord and it bent the socket in the laptop or something)
But sometime they do fix this for free, as sometimes people are new or lazy and don't both to do checks on things and may just fix it. (can't say it happens a lot, but it does happen)

id say worst case scenario is you send it in, they say it was your fault., you either pay to have it fixed, or say no you can't pay for it and they send it back to you.