There was a official reason for killin Yue and that was treason.
The exact meaning of phrase "mo xu you" was debated among history scholars. The classic translation for "mo xu you" was "don't have to exist", so Qinhui told Hanshizhong that Yue's treason "don't have to exist" in order for him to kill Yue. This version is generally discredited. The majority explanation now is that "mo xu you" was actually phrased as a sort of conformational question(as Chinese is a tonal language, that detail was not passed down in written record). If that's true, the closest english translation of the phrase would amount to something like "Surely Yue's treason is real, isn't it?!" So Qinhui was not denying Yue's committed treason, he was strongly asserting it. So in fact, Yue Fei was framed treason and killed for it, of course with the silent approval of Gaozhong.
As to why GaoZhong's so insistant in killing Yue, there are many reasons. The most important one is that Yue had become the arch enemy to his policy of making peace with Jin. Gaozhong, for reasons that needs a full paper to explain, never believed Southern Song would be able to completely conquer Jin. Therefore, in his eyes, Yue's ambition would only lead his country into a never-ending cycle of agression and retaliation. Gaozhong can't openly debate with Yue since Yue has the moral highground not to mention his reputation among military. But with Yue around, Gaozhong could never quench the hawkish faction within the government(he can't kill government officials because of a law passed down by the first Emperor of Song). So, Gaozhong has to kill Yue to successfully make peace with Jin. Finially, Gaozhong used the ultmate excuse to do that:treason. (btw, soon after Gaozhong was dead, Yue's case was reverted and Qinhui's whole family were executed, for a tool of Gaozhong, history had judged him harshly)