Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g.
N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from https://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf. 2009
or
Tammo tom Dieck. Algebraic topology. Zurich: European Mathematical Society (EMS), 2008. (7.9)
CGWH is well known to be a cartesian closed category (references above prove this). However, there appears to be two different ways to define the topology on the space TOP(X,Y) of continuous maps. Strickland topologises TOP(X,Y) with the k-ification of the topology with sub-basis $$\{f \colon X \to Y \,\, |\,\, f(u(K))\subset A\},$$ where $u\colon K \to X$ is a continuous map, where $K$ is compact Hausdorff, and $A\subset Y$ is open. In the CGWH case, and by using Lemma 1.4(b) of Strickland's paper, this coincides with the definition in May's "A concise course in Algebraic topology" chapter 5, where TOP(X,Y) is topologised with the k-ification of the topology with sub-basis: $$\{f \colon X \to Y \,\,|\,\, f(K)\subset A\},$$ where $K$ is a compact Hausdorff subset of $X$, and $A\subset Y$ is open.
However, tom Dieck topologises $TOP(X,Y)$ with the k-ification of the topology with sub-basis $$\{f \colon X \to Y \,\,|\,\, f(K)\subset A\},$$ where $K$ is a compact (not necessarily Hausdorff, it seems) subset of $X$, and $A$ is open. This is also the "official" function space topology in appendix A1 of Rudolf Fritsch and Renzo A. Piccinini. Cellular structures in topology, volume 19 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990.
These two topologies in TOP(X,Y) are clearly the same if $X$ is Hausdorff. Do they coincide in general?
EDIT: As Ivan Yudin mentioned below, given that we have the adjunction between $(\_)\times Y$ and $C(Y,\_)$, where $X \times Y$ is topologised with the k-ification of the product topology, it then follows, assuming all references (and my interpretation of them) are correct that given CGWH spaces $X$ and $Y$ the two topologies on $TOP(X,Y)$, which k-ify the topologies with sub-basis $\{f \colon X \to Y \,\,|\,\, f(K)\subset A\}$ $A\subset Y$ open, and $K \subset X$ compact, and with sub-basis $\{f \colon X \to Y \,\,|\,\, f(K)\subset A\}$ $A\subset Y$ open, and $K \subset X$ compact and Hausdorff coincide.