"At this stage our No 1 priority is getting Jimmy fully healthy and ready for the home summer."
Neesham is expected to bat today if required, as New Zealand strive to save the test.
With concerns of senior seamer Tim Southee's fitness, after he pulled up with an irritated disc in his back on the second morning of the test, it gives added cover.
Another left armer, Neil Wagner, has already been summoned as cover for Southee and will join the squad in Perth.
The 30-year-old McClenaghan has made his impact for New Zealand in the one-day game, where he's taken 76 wickets in 43 ODIs.
"Mitch has the ability to run in all day, bowl quick spells and has experience on the International stage," said Hesson.
"He'll bring a lot to the group and is obviously very excited to come and join the squad."