Charmonium is a valuable probe in heavy-ion collisions to study the properties of the quark gluon plasma, and is also an interesting probe in small collision systems to study cold nuclear matter effects, which are also present in large collision systems. With the recent observations of collective behavior of produced particles in small system collisions, measurements of the modification of charmonium in small systems have become increasingly relevant. We present the results of $J/\psi$ measurements at forward and backward rapidity in various small collision systems, $p$$+$$p$, $p$$+$Al, $p$$+$Au and $^3$He$+$Au, at $\sqrt{s_{_{NN}}}$=200 GeV. The results are presented in the form of the observable $R_{AB}$, the nuclear modification factor, a measure of the ratio of the $J/\psi$ invariant yield compared to the scaled yield in $p$$+$$p$ collisions. We examine the rapidity, transverse momentum, and collision centrality dependence of nuclear effects on $J/\psi$ production with different projectile sizes $p$ and $^3$He, and different target sizes Al and Au. The modification is found to be strongly dependent on the target size, but to be very similar for $p$$+$Au and $^{3}$He$+$Au. However, for 0%--20% central collisions at backward rapidity, the modification for $^{3}$He$+$Au is found to be smaller than that for $p$$+$Au, with a mean fit to the ratio of $0.89\pm0.03$(stat)${\pm}0.08$(syst), possibly indicating final state effects due to the larger projectile size. Comment: 362 authors, 68 institutions, 23 pages, 28 figures, 3 tables, 2014 and 2015 data. v3 is version accepted for publication in Phys. Rev. C. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.html
