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For given spaces $X$ and $Y$, let $map(X,Y)$ and $map_\ast(X,Y)$ be theunbased and based mapping spaces from $X$ to $Y$, equipped with compact-opentopology respectively. Then let $map(X,Y;f)$ and $map_\ast(X,Y;g)$ be the pathcomponent of $map(X,Y)$ containing $f$ and $map_\ast(X,Y)$ containing $g$,respectively. In this paper, we compute cohomotopy groups of suspended complexplane $\pi^{n+m}(\Sigma^n \mathbb{C} P^2)$ for $m=6,7$. Using these results, weclassify path components of the spaces $map(\Sigma^n \mathbb{C} P^2,S^m)$ up tohomotopy equivalent. We also determine the generalized Gottlieb groups$G_n(\mathbb{C} P^2,S^m)$. Finally, we compute homotopy groups of mappingspaces $map(\Sigma^n \mathbb{C}P^2,S^m;f)$ for all generators $[f]$ of$[\Sigma^n \mathbb{C} P^2,S^m]$, and Gottlieb groups of mapping componentscontaining constant map $map(\Sigma^n \mathbb{C} P^2,S^m;0)$.

Certain homotopy properties related to $\text{map}(\Sigma^n \mathbb{C} P^2,S^m)$