Sometimes a temporary cover is built first to minimise disruption on the surface.
Here's an old example of an umbrella bridge over Oxford Circus during the construction of the Victoria Line. There's a longer video out there of the construction of the Victoria Line that covers this in more detail.
Some of the stations on the Broadway Subway Skytrain extension in Vancouver use a similar approach, where half the road is closed and a road deck is built, then traffic is shifted over to the new road deck while excavation takes place from the side. There's some great views of this while riding the bus.
This is not what the cover in cut and cover refers to. This is a temporary bridge erected over a cut and cover operation. When the excavation (the "cut") and structure are completed underneath and the backfill/concrete is placed up to ground surface (the "cover") the temporary bridge will be pulled out. The "cover" part is about covering up the buried structure after it's built to put the ground surface back to where it was.
Edit: If the road deck is left in place permanently then it is a permanent elevated road deck built over a cut and cover tunnel. I can see how some people might consider this the "cover," but that is atypical in the industry and not what people are usually talking about when they say cut and cover. I'll concede that this approach sometimes happens, but I wouldn't call it "often" like GP does and I'd also note that even under this scheme the final surface/cover (e.g., the roadway) is completed after the underground excavation and structure are finished, meaning that the cut still precedes the cover.
The most common terms for this in the industry are "lid" or "cap." As in, you put the lid on, or cap, the excavation or cut and cover tunnel.
Some time ago I saw a list of all (i'm not in construcion so I only assume it was all) the different options. many people have innovated many different ways to dig while minizing surface impact. For my purposes if you eventually dig down I consider them all cut and cover - but of course if you are in construction the differences matter. For that matter even to me chose matter - but only as details that we need to argee on before starting work.
Here's an old example of an umbrella bridge over Oxford Circus during the construction of the Victoria Line. There's a longer video out there of the construction of the Victoria Line that covers this in more detail.
https://youtu.be/9ghWdnYtp2I
Some of the stations on the Broadway Subway Skytrain extension in Vancouver use a similar approach, where half the road is closed and a road deck is built, then traffic is shifted over to the new road deck while excavation takes place from the side. There's some great views of this while riding the bus.