Can I delete photos after backing up icloud

No. iCloud Photos is a syncing service. Any photo you delete from a device where iCloud Photos is turned on will also be deleted from iCloud and from any other device connected to iCloud Photos. You can potentially reduce the storage space required on a device by turning on Optimize Storage. See: https://support.apple.com/en-us/HT204264

Apple does not currently offer cloud-only storage for photos. If that is what you want or need, you may want to explore 3rd party options such as Google Photos or Dropbox.

Another way that photos may be on iCloud (if you don’t have iCloud Photos turned on) is via a backup of an iOS device. The only thing you can do with a backup is use it to restore a device or set up a new device. The contents of backups are not viewable at iCloud.com and if you delete a photo from a device it will not be included in future backups and will be lost.

If you turn on iCloud Photos on a device and also turn on the iCloud Photos Optimize Storage option, you can have smaller space-saving versions of your photos on a device with the full-size originals stored on iCloud. See the information in Set up and use iCloud Photos - Apple Support including the following from that link:

iCloud Photos keeps all your photos and videos in their original, high-resolution version. To save storage space on your device, you can turn on Optimize Storage.

If you turn on Optimize Storage, iCloud Photos automatically manages the size of your library on your device. Your original photos and videos are stored in iCloud and space-saving versions are kept on your device. Your library is optimized only when you need space, starting with the photos and videos you access least. You can download the original photos and videos over Wi-Fi or cellular when you need them.

If you turn on Download Originals, iCloud Photos keeps your original, full-resolution photos and videos in iCloud and on your device.

I have a new Apple ID which is for my small business - when I created it somehow all my wife's photo's from her Apple ID appeared. If I delete these photos from my new Id will this impact my wife's library - both of us use iCloud via our own Apple ID's?

conrad282 wrote:

I have a new Apple ID which is for my small business - when I created it somehow all my wife's photo's from her Apple ID appeared. If I delete these photos from my new Id will this impact my wife's library - both of us use iCloud via our own Apple ID's?

Sorry, I can't help with that as I don't know of any way that all photos from one Apple ID can appear on other Apple ID and I have never heard of that before. There is a Contact Support link at the top right of this page that may help.