  We present a measurement of the transverse single-spin asymmetry for $\pi^0$
and $\eta$ mesons in $p^\uparrow$$+$$p$ collisions in the pseudorapidity range
$|\eta|<0.35$ and at a center-of-mass energy of 200 GeV with the PHENIX
detector at the Relativistic Heavy Ion Collider. In comparison with previous
measurements in this kinematic region, these results have a factor of 3 smaller
uncertainties. As hadrons, $\pi^0$ and $\eta$ mesons are sensitive to both
initial- and final-state nonperturbative effects for a mix of parton flavors.
Comparisons of the differences in their transverse single-spin asymmetries have
the potential to disentangle the possible effects of strangeness, isospin, or
mass. These results can constrain the twist-3 trigluon collinear correlation
function as well as the gluon Sivers function.
