  Transverse single-spin asymmetries $A_{\textrm{N}}$ of forward neutrons at
pseudorapidities larger than 6 had only been studied in the transverse momentum
range of $p_{\textrm{T}} < 0.4$ GeV/$c$. The RHICf Collaboration has extended
the previous measurements up to 1.0 GeV/$c$ in polarized $p+p$ collisions at
$\sqrt{s}~=~510$GeV, using an electromagnetic calorimeter installed in the
zero-degree area of the STAR detector at the Relativistic Heavy Ion Collider.
The resulting $A_{\textrm{N}}$s increase in magnitude with $p_{\textrm{T}}$ in
the high longitudinal momentum fraction $x_{\textrm{F}}$ range, but reach a
plateau at lower $p_{\textrm{T}}$ for lower $x_{\textrm{F}}$. For low
transverse momenta the $A_{\textrm{N}}$s show little $x_{\textrm{F}}$
dependence and level off from intermediate values. For higher transverse
momenta the $A_{\textrm{N}}$s show also an indication to reach a plateau at
increased magnitudes. The results are consistent with previous measurements at
lower collision energies, suggesting no $\sqrt{s}$ dependence of the neutron
asymmetries. A theoretical model based on the interference of $\pi$ and $a_1$
exchange between two protons could partially reproduce the current results,
however an additional mechanism is necessary to describe the neutron
$A_{\textrm{N}}$s over the whole kinematic region measured.
