Help me map an abandoned road on public land?

The lifecycle prefix abandoned: works by prepending it to the tag's key which the feature would have had before it was abandoned. I do not know what this one was/is. Maybe a highway=service or highway=track. Depends on its use. However, is this road really abandoned, or just used by few city vehicles? It also looks like it is in a reasonable state, so I would just use the normal highway tag (not abandoned:). For example: highway=track + tracktype=grade1 + surface=asphalt. If it was a "unclassified" road before it was "abanoned" (but if it is still in a good state for such a road) you may add disused:highway=unclassified.

Regarding access tags: apparently everything except some people are forbidden to use this road. I would go for access=private. In addition, if is is well known and somehow obvious that it is tolerated by the city if someone gets around/through the gate, then also use foot=permissive + bicycle=permissive on the way. I would map it as gate - not as fence. It still can be opened by some people!

Map the gate as node of the way. Apply the same access tags to the gate as you used for the way. Maybe also add this tag: description=gate, which is an entrance to a public land, but the city is required by regulations to have this gate closed and locked in front of it. It is trivial for bikes and pedestrians to walk around the gate, although it is impassible by cars.