  In this paper we study some properties of bi-Hamiltonian deformations of
Poisson pencils of hydrodynamic type. More specifically, we are interested in
determining those structures of the fully deformed pencils that are inherited
through the interaction between structural properties of the dispersionless
pencils (in particular exactness or homogeneity) and suitable finiteness
conditions on the central invariants (like polynomiality). This approach
enables us to gain some information about each term of the deformation to all
orders in $\epsilon$. Concretely, we show that deformations of exact Poisson
pencils of hydrodynamic type with polynomial central invariants can be put, via
a Miura transformation, in a special form, that provides us with a tool to map
a fully deformed Poisson pencil with polynomial central invariants of a given
degree to a fully deformed Poisson pencil with constant central invariants to
all orders in $\epsilon$. In particular, this construction is applied to the so
called $r$-KdV-CH hierarchy that encompasses all known examples with
non-constant central invariants. As far as homogeneous Poisson pencils of
hydrodynamic type is concerned, we prove that they can also be put in a special
form, if the central invariants are homogeneous polynomials. Through this we
can compute the homogeneity degree about the tensorial component appearing in
each order in $\epsilon$, namely the coefficient of the highest order
derivative of the $\delta$.
