  Two meromorphic functions $ f $ and $ g $ are said to share a value $
s\in\mathbb{C}\cup\{\infty\} $ $ CM $ $ (IM) $ provided that $ f(z)-s $ and $
g(z)-s $ have the same set of zeros counting multiplicities (ignoring
multiplicities). We say that a meromorphic function $ f $ share $
s\in\mathbb{C} $ partially $ CM $ with a meromorphic function $ g $ if $
E(s,f)\subseteq E(s,g)$. It is easy to see that the condition ``partially
shared values $ CM $" is more general than the condition ``shared value $ CM
$". With the idea of partially shared values, in this paper, we prove some
uniqueness results between non-constant meromorphic functions and their shifts
or generalized differences. We exhibit some examples to show that the result of
{Charak \emph{et al.}} \cite{Cha & Kor & Kum-2016} is not true for $k=2 $ or $
k=3 $. We find some gaps in proof of the result of {Lin} \emph{et al.}
\cite{Lin & Lin & Wu}, and we not only correct them but also generalize their
result in a more convenient way. A number of examples have been exhibited to
validate a certain claim of the main results of this paper and also to show
that some of the conditions are sharp. In the end, we have posed some open
questions for further investigation of the main result of the paper.
