I accidentally used xkill on my desktop - how can I get it back?

The desktop icons are managed by file browser, on regular running try

$ ps ax | grep nautilus
 2464 ?        Sl     0:11 nautilus -n
 6390 pts/2    S+     0:00 grep --color=auto nautilus

From man nautilus

   -n
   --no-default-window
          Only create windows for explicitly specified URIs.

In this mode nautilus will not show a window, and it run like a daemon. So for you case use:

nautilus -n & disown

It turns out this can be fixed with Gnome Tweak Tool.

Using xkill on the desktop sometimes turns off the "Icons on Desktop" setting:

Turning this back on fixed it:

If the setting is not turned off, then just turning it off and on again brings them back.

My pronouns are He / Him

All you have to do is open Nautilus and all will be back.