laptops and chargers can be warm/hot when in use. This is not necessarily incorrect, but it is important they are both positioned with good airflow.
The 'box' in the middle of the power cord can get hot.
This is the 'charger'. It is important it is used in a position which allows maximum airflow, as it is only this that keeps it cool.
If you look on the bottom of the 'box' are there 4 tiny feet.
It is my opinion these are of little use on a hard surface, let alone a soft one such as a carpet etc., perhaps then even dangerous.
Thus I add much higher stick-on rubber feet, and when possible hang the box entirely in air.
Try sucking out any dust from the laptop's cooling vents, using a suction type hoover (distant from the laptop) with a plastic hose and nozzle.
Does the laptop itself still get hot underneath and if so, does it feel hotter than another laptop?
Is this heat where the battery is located, which presumably is a replacement for the earlier one that failed.
Depending on how long you have had it, it may be worth considering replacing the battery.
Have you tried a laptop stand from a cheapo store such as The 99p Shop (in the UK).
The plastic one with 4 legs is good, but do not twist (only slide) the two adjustable legs, otherwise you will shear the small slide pin and then must superglue these legs at the correct length.
These stands lift up the laptop, creating a good cooling position.
It might help to look inside, or have someone more computer literate do this, in case there has been a build up of dust/fluff.
As a matter of interest, where are you located? I am in Hammersmith, West London.
Please advise back on progress.
Good Luck - Keep us posted.