An $\Omega$-algebra over a field $K$ is a $K$-algebra $A$ with a set of multilinear operators $\Omega$, where $\Omega=\bigcup_{m=1}^{\infty} \Omega_{m}$ and each $\Omega_{m}$ is a set of $m$-array multilinear operators on $A$. On the other side, let consider the definition of Hom-Lie algebras as follows:
A Hom-algebra $L$ is called a Hom-Lie algebra, if $L$ is anticommutative as an algebra. i.e., $[x,y]=-[y,x]$ and the following identityholds $$ [[x,y],\alpha(z)]+[[z,x],\alpha(y)]+[[y,z],\alpha(x)]=0, $$ for any $x,y,z \in L$ and $\alpha: L \to L$ is a linear map.
Can Hom-Lie algebras be seen as a $\Omega$-algebra?
Sure, $L$ is an $\Omega$-algebra with $\Omega_1 = \{\alpha: L\to L\}$ and $\Omega_2 = \{[\cdot,\cdot] : L \times L \to L\}$ and $\Omega_m = \varnothing$ for $m>2$, i.e. with one (linear) unary operation and one (bilinear) binary operation.
Probably, Hom-Lie algebras form a variety in the sense of universal algebra.