"publicity or press agent," 1945, also as a verb by that year, said to have been coined in show biz magazine "Variety" (but this is not the first attested use), supposedly from name of Gene Flack, a movie agent, but influenced by flak. There was a Gene Flack who was an advertising executive in the U.S. during the 1940s, but he seems to have sold principally biscuits, not movies.
: The flack description is also worth quoting
: his publishers, who flack it into a best seller/ He's not flakking for ulterior motives
[origin unknown; said to be fr the name of Gene Flack, a moving-picture publicity agent, and first used in the show-business paper Variety; probably influenced by flak]