My phone won’t backup. It says I don’t have enough space but I just purchased more than my phone holds.

You don't have more storage on your phone if you buy more iCloud storage.

iCloud storage is not the same as storage on your iPhone.

You can’t buy more storage on your phone itself but here are two links that can help you to free up storage on your phone:

https://support.apple.com/en-us/HT201656

Optimizing photo storage on your iPhone can help a lot

https://support.apple.com/en-us/HT205703

I am not aware what you have purchased and what is your storage size requirement.