No, the funnel you describe sucks air from the outside to the heatsink and tbh, you can unscrew it from your sidepanel and ditch it,
your whole case is covered in dust bunnies though, time for a cleanout
and your monitor lead plugs into a socket on the I/o panel at the back yes?
You are using onboard graphics, there is no graphics card in your Pc
but for the fan yes, you can either upgrade to a better aftermarket heatsink or just replace the fan with one of the same size, either way you will need to clean the fluff and dust from the heatsink that has accumulated there,
If you google 'replace heatsink' you should get some gluetube videos showing you how to remove the hs, clean and reapply thermalpaste and refit or replace with an aftermarket like the Hyper 212 for example
Moto