  Context. White-light stellar flares are proxies for some of the most
energetic types of flares, but their triggering mechanism is still poorly
understood. As they are associated with strong X and UV emission, their study
is particularly relevant to estimate the amount of high-energy irradiation onto
the atmospheres of exoplanets, especially those in their stars' habitable zone.
Aims. We used the high-cadence, high-photometric capabilities of the CHEOPS and
TESS space telescopes to study the detailed morphology of white-light flares
occurring in a sample of 130 late-K and M stars, and compared our findings with
results obtained at a lower cadence. We developed dedicated software for this
purpose. Results. Multi-peak flares represent a significant percentage
($\gtrsim 30$\%) of the detected outburst events. Our findings suggest that
high-impulse flares are more frequent than suspected from lower-cadence data,
so that the most impactful flux levels that hit close-in exoplanets might be
more time-limited than expected. We found significant differences in the
duration distributions of single-peak and complex flare components, but not in
their peak luminosity. A statistical analysis of the flare parameter
distributions provides marginal support for their description with a log-normal
instead of a power-law function, leaving the door open to several flare
formation scenarios. We tentatively confirmed previous results about
quasi-periodic pulsations in high-cadence photometry, report the possible
detection of a pre-flare dip, and did not find hints of photometric variability
due to an undetected flare background. Conclusions. The high-cadence study of
stellar hosts might be crucial to evaluate the impact of their flares on
close-in exoplanets, as their impulsive phase emission might otherwise be
incorrectly estimated. Future telescopes such as PLATO and Ariel will help in
this respect.
