If $R$ and $S$ are binary relations, then the composition relation $R\circ S$ is usually defined by $a\mathrel{R\circ S} c$ if and only if there is $b$ such that $a\mathrel{R}b$ and $b\mathrel{S}c$.
In the special case where the relations are the (graphs) of functions $f$ and $g$, this produces the (graph) of the usual composition function $f\circ g$, since $(f\circ g)(x)=z\iff \exists y\ f(y)=z$ and $g(x)=y$.
In the case of an order $\lt$, what the relation $\lt^2$ would mean is $\lt\circ\lt$, which would be defined by $a\lt^2 c$ if and only if there is $b$ such that $a\lt b\lt c$.
In a discrete order, such as the usual order $\lt$ on the integers $\mathbb{Z}$, this means that $r\mathrel{\lt^2} t$ if and only if there is $s$ with $r\lt s\lt t$, which is the same relation as $r+2\leq t$, which may explain the confusing $+2$ issue you mention.
Meanwhile, in a reflexive relation $\leq$, we have $\leq^2=\leq$, since $a\leq b\iff a\leq a\leq a\leq b$.