According to director Ridley Scott, there are very few deleted scenes from Alien: Covenant. It may not have set the box office alight and received divisive reviews, but Ridley Scott's return to the sci-fi horror franchise he birthed in 1979 is nothing if not ambitious, serving as both a prequel to Alien that explains the origins of the xenomorph and a sequel to previous outing Prometheus that extends its grand themes of creation.
There is a lot of Covenant that audiences saw before release that wasn't in the film, however. Two prologues - one detailing the crew of the eponymous vessel before cryosleep and the other detailing what happened to the survivors of Prometheus between movies - used extensive fresh footage (including proper appearances from James Franco and Noomi Rapace respectively) and several trailers focused more on the human heroes, specifically Katherine Waterston's Daniels on the Covenant before discovering the horrors of the Engineer homeworld. These apparently aren't deleted scenes, though.
Speaking with Collider, Ridley Scott discussed his editing process and during the conversation stated that there weren't many deleted scenes from the film:
Ridley Scott: The only reason I occasionally take my hat off to screenings is you think you know everything, but you know you don't. Something inordinately simple you've assumed everyone understands and they don't. That's the key thing, isn't it? You know when it's a bit too long. And also, if you're a bit too long you then ask "Right speed? Too slow? Too fast?" If they go, "no, too fast" you go "sh*t, I've got to slow down" - because as you edit, you can get weary and want to cut, cut, cut.
Collider: Oh, so were there a lot of deleted scenes?
Michael Fassbender: I don't think so.
Scott: I always come out at almost 2:23 first cut, look at it, and go "I've got to get to two hours".
Alien: Covenant as released comes in at two hours, three minutes. That does of course mean that around twenty minutes of footage was lost between Scott's initial edit and the finished product, but from what he says about his process, particularly in reference to test screenings and how they altered pace, it was mostly existing scenes being trimmed rather than entire swathes of content being cut. This is a similar situation to Rogue One; despite extensive reshoots, the Star Wars Story reportedly doesn't have much in the way of full, completed scenes that were removed.
These comments would seem to indicate that all the marketing footage was only ever intended to be used in trailers. This has been the prevailing explanation and would be similar to Prometheus, which constructed an entire TED talk by Guy Pearce as Peter Weyland to drum up hype. Although it is possible that there were reconstituted from cut footage and only don't count as "deleted" in the director's mind because Fox found a later purpose from them. Indeed, if Scott was forced to cut ten plus minutes of his film, the banter between the crew would be an obvious candidate to go.
And even if the scenes are just trimmed, there's the possibility of more content in those slight chops; Prometheus had several "alternate" scenes that better explained the Engineer's sacrifice in the opening and what Weyland wanted from his creator in the climax.
In the past year, there's been a rise of high-profile "Extended Cuts" that "improve" the theatrical cuts, most prominently with the DCEU, and we're now seeing a movement in the other direction, with a lack of deleted scenes viewed as indicating a purer directorial vision. The most recent case is Wonder Woman, which reportedly has no cut scenes at all. For Covenant, though, we'll have to wait for the home release to see if anything of substance comes from what Scott removed.