A common query we get is 'can I paint over my inkjet prints'? Either with paint, or inks, or a finishing medium like gel medium or varnish.
The short answer is yes, you can, and our clients have used both acrylics and oils with success, as well as a variety of other mediums. We'd generally recommend acrylics as our pigment inks are water based, and thus it's probably better - especially if painting soon after printing - to stick to water based materials.
The longer answer is that, like all things in art, it's really a process you have to test to see the results - there's no guarantee a particular paint/ink/whatever will work nicely on top of inkjet coated paper - you just have to try! Have an A4 sample print made and give it a go - usually it will work, but sometimes odd reactions can occur and your medium won't adhere properly to the surface of the print, or you will find the surface to have too much 'traction' when working the medium.
There's also no guarantee about the archival longevity of your piece - funny chemical reactions between mediums can occur, and your particular combination probably won't have been tested (but in reality in art, most things aren't tested!). So you might have to accept that over time you might get yellowing or similar - but odds are that the natural ageing will be a visually enjoyable part of the whole process, so best to embrace this rather than fear it!