A Lie \(p\)-algebra \(L\) is called \(n\)-power closed if in every section of \(L\), any sum of two \(p^{i+n}\)-th powers is a \(p^i\)-th power \((i>0)\). The authors prove that if \(L\) is residually nilpotent and \(n\)-power closed for some \(n\geq 0\) then \(L\) is \((3p^{n+2}+1)\)-Engel if \(p\geq 2\) and \((3\cdot 2^{n+3}+1)\)-Engel if \(p=2\). They also show that any associative algebra \(R\) generated by nilpotent elements satisfies an identity of the form \((x+y)^{p^n} = x^{p^n} +y^{p^n}\) for some \(n\geq 1\) if and only if \(R\) satisfies the Engel condition.