Is there a polynomial $g(x)$ such that $p \circ f=g \circ p$, where $f(x)=9x+30x^3+27x^5+9x^7+{{x}^{9}}$?
$$ \begin{array}{rcl} p(x)&=&9x,\\ g(x)&=&\displaystyle \frac{x^9}{43046721}+\frac{x^7}{59049}+\frac{x^5}{243}+\frac{10 x^3}{27}+9 x. \end{array} $$
Upd. For the new version $$ \begin{array}{rcl} p(x)&=&x^2,\\ g(x)&=&x (x+3)^2 \left(x (x+3)^2+3\right)^2; \end{array} $$
$$ \begin{array}{rcl} p(x)&=&x^4+4 x^2,\\ g(x)&=&x (x+3)^2 \left(x (x+3)^2+3\right)^2. \end{array} $$
$$ \begin{array}{rcl} p(x)&=&x^2 \left(x^2+3\right)^2,\\ g(x)&=&x (x+3)^2 \left(x (x+3)^2+3\right)^2. \end{array} $$