Klate sustained a groin injury in the Nedbank Cup last-16 loss to Jomo Cosmos on April 4.
Hunt is also missing the services of teen striker Phakamani Mahlambi and January signing Granwald Scott.
"Daine is still injured," Hunt told KickOff.com.
"So he will not be available for the upcoming matches. But we are missing a lot of good players. We are missing Phakamani Mahlambi and Granwald Scott as well."
Asked how Wits will cope without their casualties, Hunt said: "I don't know."
This website, however, understands defender Nazeer Allie and midfielder Xola Mlambo are back at training following a stint on the sidelines.
The title contenders next host Golden Arrows at Bidvest Stadium tomorrow (19h30).
Hunt's side are second on the log with 41 points from 20 games - a point behind Cape Town City, but with two matches in hand.