Among the results of the paper we have the following proposition. Let $k>1,s\geq1,n=k\cdot s+1$ and let $(Q,A)$ be an $n$-groupoid. Then, $(Q,A)$ is an near-P-polyagroup (briefly: NP-polyagroup) of the type $(s,n-1)\textit{[}:[11],1.3\textit{]}$ iff the following statements hold: $(i)$ $(Q,A)$ is an $\langle1,n\rangle$-and $\langle1,s+1\rangle$-associative $n$-groupoid extit{[}extbf{or} $\langle1,n\rangle$-and $\langle(k-1)\cdot s+1,k\cdot s+1\rangle$-associative $n$-groupoidextit{]}; and $(ii)$ for every $a^n_1\in Q$ there is extbf{at least one} $x\in Q$ and extbf{at least one} $y\in Q$ such that the following equalities hold $A(a^{n-1}_1,x)=a_n$ and $A(y,a^{n-1}_1)=a_n$.