On the print, underneath the image area, I want to add a number.
In Photoshop, I can increase the Canvas Size (coloured white) and the use the Type Tool to create the number.
But, this means that the shade of white will have to be printed too. Right?
No, pure White in the image is simply no ink, leaving whatever the paper color is. This assumes the Paper profile doesn't force some color on pure white image areas. Most printers do not print with white ink.
If it does force some color on white areas, you can try and create a larger transparent image layer where you paste a copy of your flattened image on. Then between the transparent bottom layer and you image top layer you can create a text layer. However, there is a chance that the printer driver will change the transparent area to White though, so you should try it first.