With the caveat that I'm sure it varies based on publisher, genre, etc., but -- generally, if you were an author who was published and the book cover was designed by the publisher, but you wanted to do marketing on your own -- would you be able to use that book cover image on a personal website, in paid ads on social media, on flyers, etc.? Would you need to pay a licensing fee?
Your contract should spell out what, if any, rights you have to use the cover art and/or book excerpts for promotional purposes. Typically, smaller or independent presses will be more than happy to add in those rights during negotiation since they will typically have a much smaller budget for promotion and will rely on the author to do much of the promotional work themselves.
Larger presses, on the other hand, usually have a large marketing department that will handle all of the initial press push for the book and will be less likely to want the author doing any additional promotion that may go against their particular policies or strategies.
Typically, "no". Not unless there's a contract or license that allows it. The cover was commissioned and paid for by the publisher.
However, non-deceptive use for the purpose of selling more books would probably be happily approved by the publisher.