  For a natural number $m$, generalized $m$-gonal numbers are defined by the
formula $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this
paper, we determine a criterion on $a,b,c,m$ for which the weighted ternary sum
$P_{a,b,c,m}:=ap_m(x)+bp_m(y)+cp_m(z)$ is almost universal. We also prove for
some $a,b,c,m$ that the form $P_{a,b,c,m}$ is not almost universal, while it
represents all possible congruence classes.
