  A finite morphism $f:X\to \mathbb P^2$ of a a smooth irreducible projective
surface $X$ is called an almost generic cover if for each point $p\in \mathbb
P^2$ the fibre $f^{-1}(p)$ is supported at least on $deg(f)-2$ distinct points
and $f$ is ramified with multiplicity two at a generic point of its
ramification locus $R$. In the article, the singular points of the branch curve
$B\subset\mathbb P^2$ of an almost generic cover are investigated and main
invariants of the covering surface $X$ are calculated in terms of invariants of
the curve $B$.
