  $(\mu;\nu)$-Hankel operators between separable Hilbert spaces were introduced
and studied recently (\textit{$\mu$-Hankel operators on Hilbert spaces},
Opuscula Math., \textbf{41} (2021), 881--899). This paper, is devoted to
generalization of $(\mu;\nu)$-Hankel operators to the (non-separable in
general) case of Hardy spaces over compact and connected Abelian groups. In
this setting bounded $(\mu;\nu)$-Hankel operators are fully described under
some natural conditions. Examples of integral operators are considered.
