  I present a class of theories that generalize quasilinear MOND (QUMOND). Like
QUMOND, these GQUMOND theories require solving only the linear Poisson equation
(twice). Unlike QUMOND, their Lagrangian depends on higher derivatives of the
Newtonian potential. They thus dictate different "phantom" densities as virtual
sources in the Poisson equation for the MOND potential. These theories might
open new avenues to more fundamental theories, and have much heuristic value. I
use them to demonstrate that even within limited classes of modified-gravity
formulations of MOND, theories can differ substantially on lower-tier MOND
predictions. Such GQUMOND theories force, generically, the introduction of
dimensioned constants other than the MOND acceleration, $a_0$, such as a
length, a frequency, etc. As a result, some of these theories reduce to QUMOND
itself only, e.g., on length scales (or, in other versions, dynamical times)
larger than some critical value. But in smaller systems (or, alternatively, in
ones with shorter dynamical times), MOND effects are screened, even if their
internal accelerations are smaller than $a_0$. In such theories it is possible
that MOND (expressed as QUMOND) applies on galactic scales, but its departures
from Newtonian dynamics are substantially suppressed in some subgalactic
systems -- such as binary stars, and open, or globular star clusters. The same
holds for the effect of the galactic field on dynamics in the inner solar
system, which can be greatly suppressed compared with what QUMOND predicts.
Tidal effects of a galaxy on smaller subsystems are the same as in QUMOND, for
the examples I consider. I also describe briefly versions that do not involve
dimensioned constants other than $a_0$, and yet differ from QUMOND in important
ways.
