Is there a degree $5$ homogeneous polynomial $f \in \mathbb{C}[x_0,\dots,x_4]$ such that $Z(f) \subset \mathbb{CP}^4$ is smooth apart from a single singular point, which is a node? (Node is another word for ordinary double point for me.)
The conifold $x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-5x_0 x_1 x_2 x_3 x_4$ has $125$ nodes (e.g. Example 6.15 here). The example here has $16$ nodes: https://mathoverflow.net/questions/185333/resolving-nodes-of-a-quintic-cy-3-fold .
Let $P \in \mathbb{P}^4$ be a point, let $X := \mathrm{Bl}_P(\mathbb{P}^4)$ be the blowup, let $E \subset X$ be its exceptional divisor, and let $H$ be the pullback to $X$ of the hyperplane class of $\mathbb{P}^4$. Then the line bundle $\mathcal{O}_X(5H - 2E)$ is globally generated, hence a general divisor $D \in |5H - 2E|$ is smooth by Bertini's theorem. Moreover, the map $$ H^0(X, \mathcal{O}_X(5H - 2E)) \to H^0(E, \mathcal{O}_E(5H - 2E)) = H^0(\mathbb{P}^3, \mathcal{O}_{\mathbb{P}^3}(2)) $$ is surjective, therefore a general $D$ intersects $E$ along a smooth quadric surface, hence the image of $D$ in $\mathbb{P}^4$ has a single node at $P$.