Start a process in Elisp
You can remove the let, as you only reference f once.

I'd also make it a named function; this makes it easier for you to update the definition, or to bind it to additional keys:

(defun start-mpv--file-at-point ()
  (interactive)
  (start-process "test" "output" "mpv" (ffap-file-at-point)))

(global-set-key [f3] 'start-mpv--file-at-point)


You might also want to look at the implementation of shell-command for inspiration.  Some things it does that might interest you:


killing any process that's already writing to the output buffer
erasing the current contents of that buffer
setting the comint output filter to handle carriage motion
making the output buffer visible