URL problem with WordPress site

Even though I have been designing quite a few websites in html and css I am fairly new to WordPress. I have a video guitar lessons website (Playgtr.net) that I have moved to WordPress to allow comments and more interactivity. The hosting plan I have has as main domain name ‘mrgcentral.com’ and playgtr.net is a secondary domain name (not a redirect or a forward, though). In the old website the partition of the hosting space corresponding to PlayGTR.net was ‘mrgcentral.com/playgtr’. Now I have installed wordpress on ‘mrgcentral.com/playgtr-wp’ and, after getting all the pages and posts sorted, moved the domain name ‘playgtr.net’ to that partition. Unfortunately now I have the problem that once I click on any link on the website, except for the beginning homepage where the URL shows ‘playgtr.net’ all the other pages, posts and links have to be directed to ‘mrgcentral.com/playgtr-wp’ (and it will show on the URL bar). As an example, if I click to go to the ‘contact’ page, the url will show ‘https://www.mrgcentral.com/playgtr-wp/contact/’ and not ‘https://www.playgtr.net/contact’. Should I have installed WordPress on the Playgtr.net partition to begin with? Is there something I can do now to move the whole finished WordPress site without starting again from scratch? Maybe backing up the installation and database, installing a new one on the playgtr.net partition and overwriting it with the backed up site??
I hope I have been clear enough in explaining the problem…I hope somebody will have a simple answer to this!

Thank you in advance!!

Gianni