Rings in which all elements are the sum of a central element and an element from $\mathrm{\Delta }(R)$

We define and consider in-depth the so-called $C\mathrm{\Delta }$ rings as those rings $R$ whose elements are a sum of an element in $C(R)$ and of an element in $\mathrm{\Delta }(R)$. Our achieved results somewhat strengthen these recently obtained by Ma-Wang-Leroy in Czechoslovak Math. J. (2024) as well as these due to Kurtulmaz-Halicioglu-Harmanci-Chen in Bull. Belg. Math. Soc. Simon Stevin (2019). Specifically, we succeeded to establish that exchange $C\mathrm{\Delta }$ rings are always clean as well as that exchange CN rings are strongly clean. Likewise, we prove that, for any ring $R$, the ring of formal power series $R[[x]]$ over $R$ is $C\mathrm{\Delta }$ if, and only if, so is $R$. And, furthermore, we show that, for any ring $R$, if the polynomial ring $R[x]$ is a $C\mathrm{\Delta }$ ring, then $R$ satisfies the Köthe conjecture. Some other closely related things concerning certain extensions of $C\mathrm{\Delta }$ rings are also presented.