Why do the Klingons jump between Klingon and English?

This is almost certainly a translation convention (warning: TVTropes) at work. The characters are speaking the Klingon language, but it's being translated to english for the audience's benefit.

The TVTropes page mentions this specific technique:

In some cases, the actors will begin speaking in the characters' native language, then perform a switch-over to English. This is usually accompanied by some sort of camera-move to cue the audience in to the fact that the characters should still be assumed to be speaking in their native language.

The point is to establish that the characters are speaking a foreign language, then relieve the audience of having to read subtitles for the complete scene. It also relieves the actors from having to perform the entire scene in a language that they may not know.

The translation conventions page contains a specific note about this film:

Star Trek III: The Search for Spock has Klingon commander Kruge and his officers, Torg and Maltz, confer in English, but Kruge speaking in Klingon to the rest of his crew. Though whether the convention applies in full or they're actually speaking English to conceal their conversation from the non-English-speaking crew is never made clear.

And the Star Trek III: The Search for Spock page contains a couple more notes about the Klingon dialog:

Aliens Speaking English: Although the Klingons are shown speaking their own language at first, for simplicity's sake they speak English to each other for the majority of the film. Only Kruge, Torg and Maltz are shown to actually be able to speak English though. This becomes a brief plot point when the Klingon troopers board the Enterprise and don't recognise the computer counting down from 9...8...7...6...5...4...3...2...1...

Translation Convention: Bizarrely, the scenes with the Klingons only use it on about half the lines. The Klingon Dictionary notes that it was fashion for a time for Klingon officers to speak English to each other when discussing things they didn't want their subordinates to hear.

Short answer:

The audience is supposed to assume that the fictional Klingon characters continue to speak Klingonese for the entire duration of those scenes, but the movie production shifts from using subtitles to interpret their dialog to dubbing in English language dialog in the midst of those scenes.

Long answer:

You can assume that the fictional Klingon characters actually speak Klingon all the time in every scene except when talking to non Klingons when they might have to use a non Klingon common language, so it is an artistic (or possibly financial) choice by the director whether the actors portraying the Klingon characters speak Klingon or English in any particular sentence.

So possibly the director decided to start a scene with a few lines of the Klingon characters speaking Klingon to each other with subtitles in English, and then switch to having the Klingon characters speak English for the rest of the scene in order that English-speaking audiences don't have to read subtitles fast for the entire scene.

Possibly the Human actors portraying the Klingon characters do multiple takes of every shot, sometimes speaking their lines in Klingonese and sometimes in English. Later in post production two versions of the scene were assembled, one with every line in Klingonese and one with every line in English. So the director, Leonard Nimoy, could choose which combination of Klingon lines and English lines to use in that scene by combining shots from both language versions of the scene.

Or possibly the director and screen writers had already decided with lines would be in Klingonese and which in English in the completed film and they only filmed lines of dialogue in one language.

Note that when characters say things in languages foreign to the audience of the film, there are three options:

One, leave the foreign dialog untranslated expected the audience to get a general understanding of the meaning from the action.

Two, use subtitles to translate foreign dialog.

Three, dub in translated dialog spoken by the original language actors or by other actors who speak the language being translated into better.

So when Klingon characters in Star Trek III: The Search for Spock seem to switch from speaking Klingonese to speaking English in the middle of a scene, one can assume that "really" only the Human actors switch from speaking Klingonese to speaking English, and that the Klingon characters continue to speak Klingonese, but the movie production switches from using English subtitles to using English language dubbing.