I'm not sure that I can offer a solution, but I can offer a simplification of the inequalities.

Subtracting the second inequality from the first and also keeping the second one is an equivalent set of inequalities: $-u''/u + (u'/u) w' \ge -C$, $w'' - (u'/u) w' \ge 0$. A little bit of manipulation then shows that this system of inequalities can be rewritten as
$$
\begin{cases}
-\frac{e^w}{u}\left(e^{-w}u'\right)' \ge -C , \\
u (w'/u)' \ge 0 .
\end{cases}
$$

If you make assumptions about signs of the multiplicative factors, you can move those to the other side of the inequalities and then integrate. Since (definite) integrals preserve inequalities, you get new inequalities that no longer involve derivatives of $u$ and $w$. Perhaps some version of those can help you make a conclusion about the existence or non-existence of a desired solution.

**Update**: Actually, I feel rather silly about part of what I wrote. One cannot subtract inequalities as I did above ($a \ge b$ and $c \ge d$ does not imply $a-c \ge b-d$). So the second inequality that I wrote is not actually valid. Apologies for any confusion! I'll leave what I wrote as is, though, in case the algebraic manipulations do prove useful.