Can I fix my laptop charging port?

This is a common laptop problem, all workshops should be able to repair/replace the charging port. It isn't usually a DIY job as the laptop has to be completely stripped to gain access. Unless you have stripped laptops before and you have a replacement charge port, best leave to the pros.

https://compupartsolutions.com/shop/dc-jack/5c10g89487-lenovo-g70-80-dc-in-jack/

Be wary about the cost of repair. It is about a $10 to $20 cable in aftermarket and about $60 labor or do it yourself
Lenovo 5c10g89487 cable

Here is a cheap and easy fix first unplug the battery and the power cord now take a tiny eyeglass screwdriver and inside the charging port you will see two pins Bend them toward the center pin a little bit so that the charging port is tight again no remove the back of your laptop not very hard there's a few screws in the back comes right off find the charging port that's Wiggly push it back into position and put one little drop of Gorilla Glue on the outside plastic part making sure you don't get any at all inside of the charging port leave it alone till the next day and bingo your charging port will be solid as a rock again

Might be a small wire inside the port has become loose or broken, if so it just needs some soldering

Looks like you may be in luck, its on a push in cable. However dismantling any laptop can be anywhere from very easy to very tricky and how it goes back together is well about the same.