What should I do to expand my knowledge so that I don't get stuck with such amateur questions in the future? :) I am more inclined towards an articled option, 'cause I believe we can count delays, although delays are abstract in the sense that they're not physical objects... The context is: "Your message may arrive with a bit of (a) delay."
This is a "six of one, half-a-dozen of the other" usage. My default is to include the article, but both versions are perfectly common...
There's no significant difference between AmE and BrE, and I can't see any possibility of a difference in meaning. But obviously mine and OP's preferred version is in the ascendant here.
In various kinds of processing, adding "a bit of delay" would entail making an existing step in the process take slightly longer than it otherwise would have, while adding "as bit of a delay" would involve adding a new step whose purpose is to defer later actions.
If one imagines a plane approaching an airport which has a plane on the runway awaiting takeoff, an arriving pilot who reduces airspeed would be adding "a bit of delay", while one who circled would be adding "a bit of a delay". Looking at the former pilot's trajectory, no identifiable part of it would be "the added delay", but the latter pilot's path would have a circle which, if omitted, would have allowed the pilot to arrive sooner.