As a hybrid work of lyric poetry, critical prose, and narrative prose, its obvious (if distant) engenderers are Dante's self-annotated lyric sequence La Vita Nuova, and a tradition of courtly prose romances with embedded verse which stretches from Boccaccio to (seven years after Gascoigne) Sidney's Arcadia. What distinguishes Gascoigne's work is the combination of autobiography and distance.
Dante's prose told the same high (and higher all the time) flown story as his verse. Courtly romances also assumed an exact correspondence between what's expressed and the feelings of the expresser: the good poet is the noble gent.
In Gascoigne, lyric and narrative poses are set at odds. His love poetry consists of the usual abstract praise (in fact, the utilitarian benefits of Petrarchian non-specificity are specifically mocked), while his prose describes a depressingly tawdry story of pointless -- almost rote -- seduction, betrayal, rape, and epidemic hypochondria. Earlier courtly romances toed the party line, whereas Gascoigne gives the game away: Poetry is a a social tool, to be used like any other. By writing the poems, "F. J." hopes to win sex; by publishing them, Gascoigne hopes to win career advancement.
Gascoigne's poems are the ostensible point of the publication, but rather than being annotated by their author, they're embedded in a gossipy third-person narrative, and a framing fiction of piracy emphasizes both the authenticity of the account and the third-person-ness of the narrator. Meanwhile, there's no doubt that F. J. himself is the final POV arbiter, given evidence like the outburst of invective on introduction of his hated rival, the repellent smugness toward his unluckily hospitable host, and the callous cluelessness regarding the rape.... The narrator "G. T." was probably invented as a not-so-obviously-narcissistic mouthpiece to praise "F. J."'s poetry. A side effect of this third person was a new acidity in self-portraiture.
I don't want to overstate the case. "The Adventures of Master F. J." may be the first recognizable precursor of Stendhal and Flaubert, but it couldn't be mistaken for either of them. While the dialog is sometimes startlingly sharp, the narrative per se limps along on a very limited vocabulary (which seems to some extent intended: in the 1575 edition, Gascoigne hopes "that it shall be apparent I have rather regard to make our native language commendable in itself than gay with the feathers of strange birds"). Whether playing braggart or gossip, the narrator has little need to concern himself with context, and we're given no notion of how the characters live, what they wear, what they look like, their surroundings, or even their exact relationships (although the "riding tale of Bartello" is slightly clearer on that score).
There's still plenty of case to state, though, including perhaps the first appearance in a romance of the just-good-friends confidante, accompanied by what remains a supremely brutal response to "What's she got that I haven't got?" (Answer: Easy access.)
Gascoigne's work had no direct progeny, although it's easy to imagine an alternate version of literary history in which (to take an easy case) Byron would have been familiar enough with Gascoigne to use him as a model, starting an English line of mixed-genre realism. And then from Byron to Pushkin....
One thing that made this only an alternate history was the hostile response to the first edition of Gascoigne's collection, and his reaction.
The Adventures of Master F. J. by George Gascoigne, 1573