  In this paper, first we obtain an explicit formula for an outer commutator
multiplier of nilpotent products of cyclic groups with respect to the variety
$[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, $\mathfrak{N}_{c}M(\mathbb{Z}\st{n}*
\mathbb{Z}\st{n}* ... \st{n}* \mathbb{Z}\st{n}* \mathbb{Z}_{r_1}\st{n}*
\mathbb{Z}_{r_2}\st{n}* ... \st{n}* \mathbb{Z}_{r_t})$ where $r_{i+1}\mid r_i \
\ (1\leq i\leq t-1)$, $c_1+c_2+1\geq n$, $2c_2-c_1>2n-2$ and $(p,r_1)=1$ for
all prime less than or equal $c_1+c_2+n$, second we give a necessary condition
for these groups to be $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable.
