  For finite-dimensional Hopf algebras, their classification in characteristic
$0$ (e.g. over $\mathbb{C}$) has been investigated for decades with many
fruitful results, but their structures in positive characteristic have remained
elusive. In this paper, working over an algebraically closed field $\mathbf{k}$
of prime characteristic $p$, we introduce the concept, called Primitive
Deformation, to provide a structured technique to classify certain
finite-dimensional connected Hopf algebras which are almost primitively
generated; that is, these connected Hopf algebras are $p^{n+1}$-dimensional,
whose primitive spaces are abelian restricted Lie algebras of dimension $n$. We
illustrate this technique for the case $n=2$. Together with our preceding
results in arXiv:1309.0286, we provide a complete classification of
$p^3$-dimensional connected Hopf algebras over $\mathbf{k}$ of characteristic
$p>2$.
