Is there a known extension of Dirac's / Ghoulia-Houri's theorems for $k$-path existence? In the well studied problem of Hamiltonicity, several papers/theorems gave sufficient "degree conditions" for the existence of Hamiltonian path in a graph.

These include:


Dirac's theorem , 1952, which states that "A simple (undirected) graph with $n$ vertices ($n \geq 3$) is Hamiltonian if every vertex has degree $\frac{n}{2}$ or greater".
Ghouila-Houiri 's theorem, ("Une condition suffisante d’existence d'un circuit hamiltonien",1960), which generalized Dirac's result to directed graphs, showing that if $\delta^+(G),\delta^-(G) \geq \frac{n}{2}$ then G is Hamiltonian, where $\delta^+(G)$ is the minimal out degree in $G$, and by $\delta^-(G)$ the minimal in-degree.
A theorem from Bang-Jensen and Gutin's book gave a stronger result, which imply that if $\forall v\in V: d_{in}(v)+d_{out}(v)\geq n$ then the graph is Hamiltonian.




The $k$-path problem is a simple generalization of Hamiltonicity asking whether a simple path on $k$ nodes exist in a graph.


  What would be the weakest sufficient "degree conditions" from a graph such that it is bound to contain a $k$-path?
I don't know about the weakest possible conditions, but if a graph has a subgraph with minimum degree $k-1$ (that is, if its degeneracy is at least $k-1$) then a greedy algorithm can easily find a path of $k$ nodes starting at any node. The example of $K_{k-1}$, which has no $k$-path and has minimum degree $k-2$, shows that this is tight.