  Motivated by the Novikov equation and its peakon problem, we propose a new
mixed type Hermite--Pad\'{e} approximation whose unique solution is a sequence
of polynomials constructed with the help of Pfaffians. These polynomials belong
to the family of recently proposed partial-skew-orthogonal polynomials. The
relevance of partial-skew-orthogonal polynomials is especially visible in the
approximation problem germane to the Novikov peakon problem so that we obtain
explicit inverse formulae in terms of Pfaffians by reformulating the inverse
spectral problem for the Novikov multipeakons. Furthermore, we investigate two
Hermite--Pad\'{e} approximations for the related spectral problem of the
discrete dual cubic string, and show that these approximation problems can also
be solved in terms of partial-skew-orthogonal polynomials and nonsymmetric
Cauchy biorthogonal polynomials. This formulation results in a new
correspondence among several integrable lattices.
