yes, the Thunderstorm connection is possible - ongoing research work suggests that many thunderstrikes are triggered ('enabled' might be a better choice of word) by incoming galactic + solar protons - as you say, the sprites are particle/energy jets apparently accelerating back out, but there seems to be a lot more work to do on simply thunder - before Sporadic E is added to the scientific questions! The actual enabling of lightning by incoming radiation is not yet quantified.
My first worry is that Es mirrors probably happen around 100 km up, whilst a lot of the thunderstorn physics is inferred to happen between 10 - 20 km above the ground (the emission spectra of Thunderstorm "intense fluxes of electrons, gamma radiation and secondary neutrons correlated with thunderstorms/lightning" seem to be only 20% as high as Es) from this next 2018 paper. There are however some elements of thunderstorms that appear to be located at 500 km altitude...
looking on https://scholar.google.co.uk gives quite a lot of reading, I'll just choose one recent paper, for this short reply
[ref:Surveys in Geophysics, September 2018, Volume 39, Issue 5, pp 861–899 "Lightning Discharges, Cosmic Rays and Climate", Sanjay Kumar, Devendraa Siingh of Indian Institute of Tropical Meteorology Pune & BHU Varanasi]
(might be ashamedly paywalled so some 'review' quotes follow from this long and interesting read)
"Gurevich and Zybin (2001, 2005) proposed the runaway breakdown mechanism operating at a lower threshold voltage (~ 2.16 kV cm−1) which involved the passage of high-energy particles (cosmic rays) through the thundercloud" (it was previously thought that a 23 kV cm−1 electric field was needed to trigger the lightning flash)
some other keywords from just this paper: particle nucleation & cosmic rays, thunderstorm electrification & cosmic rays, triggering of lightning by cosmic rays, high-energy radiation & thunderstorm ground enhancements (TGEs), terrestrial gamma ray flashes (TGFs), the global electric circuit & climate, atmospheric general circulation lightning & climate, cloud condensation nuclei (CCN), terrestrial electron beams (TEBs), Sprites= transient luminous events (TLEs), gigantic jet (GJ), etc...loads more acronyms available
The thunderstorms apparently even accelerate muons (downwards) and might have energetic electron interactions. Physicists can measure "intense fluxes of electrons, gamma radiation and secondary neutrons correlated with thunderstorms/lightning", and " the observed spectra were consistent with the simulation results when the source region was considered at 21 km altitude and below and the derived spectrum was inconsistent for sources above 21 km altitudes. This supports the theory that TGFs are produced either inside a thundercloud or just above a thundercloud, and anyway they are not associated with high-altitude discharges (sprites)"
CERN is experimenting with trigger rates in CLOUDs (Cosmics Leaving Outdoor Droplets)
It would be nice to add Es to this area of climate research, directly or indirectly as lower atmosphere plasmas, and cosmic induced & other ion clouds are mentioned by the paper, some of which get 'scooped-up' by a forming thunderstorm -
Q: so does Es preceed the series of storms,
Q: or is Es as a result of the *lots of physics* happening during a typical storm,
so much is still apparently unknown.
this above 2018 paper has a nice (simplified) thunderstorm model here