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 snl24.com/kickoff.
"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.