iCloud Photos is a syncing service, not an external storage for your photos. It is not possible to remove your photos completely from your iPhone withiCloud Photos.
The main purpose of iCloud Photos is to store your photos and videos in iCloud as a central storage to keep your Photos Library identical and in sync across all your Apple devices. Whatever you modify in Photos on one of your devices will sync to iCloud and from there to all your other devices. So the photo and videos will be stored in iCloud, but also mirrored on your iPhone and other devices by local shadow copies. You can use iCloud Photo to save some storage on your iPhone, if you enable "Optimize iPhone Storage" in the iCloud Settings. Do you have "optimize Storage" enabled?
The Photos will replace the high resolution originals on your iPhone by smaller, optimized versions for browsing, but you cannot get the photos completely off your iPhone. You will need at least enough storage for the optimized versions. The "optimized" shadow copies may be listed as "Other" storage or "System data", depending on your system version. If your iPhone is storage is full and cannot even hold all the optimized versions, there is no point in buying more iCloud Storage. Then the best strategy would be to remove photos from iCloud Photos and save to save them elsewhere, for example on a computer. Keep only photos in iCloud Photos, that you want to see on all your devices and will fit optimized on your device with the smallest storage. As a rule of thump - the optimized versions may need 10% to 20% of the storage used in iCloud Photos for your photos. So, if you only can afford to use 10GB for photos on your iPhone, do not keep more than 50GB of photos and videos in iCloud.
