I had the same problem with a portrait photo of a black dog. When “cut” in iOS 16 the image has a stark white background - if using this as a wall paper the text in the title bar remains white and can not be read!
As has been suggested there are many apps out there that will solve the issue but although “free” have limited functionality often with poor resolution and are geared up for in app purchases! Using the inbuilt iOS features I have arrived at a workaround using the in built iOS features - a bit cumbersome but works for a one off.
Cut and copy the image.
Open Pages and make a coloured “paper” to paste the image onto thus:
Select a blank document.
Select “shape” - fourth tool from left in title bar and select eg the square and resize as suitable.
now select the “brush” tool - third from right.
Select “Fill” -> colour -> choose required background (paper) colour from the pallet.
Paste cut image onto this with necessary adjustments!
Will not save to Photos from Pages so do a Screen Shot!
GOOD TO GO!