Should there be a difference between havn and havnen when read aloud? I can't perceive any.
"Havn" sounds to me pretty much just like "hown" or "houn" would in English. "Havnen" sounds more like "how" + the "n" sound, with a bit of a glottal stop before the "n" sound.
It is almost the same in German: Hafen (harbour) and Flughafen (flight-harbour = airport).