This is a genuinely difficult shooting situation: low light, subject/camera movement, and fog all reduce sharpness and make autofocus struggle. Some softness is unavoidable because haze lowers contrast and you’re literally shooting through it.

Best options:

• If this is for publicity/documentation, ask the lighting director/director whether you can shoot a dress rehearsal with more light, then underexpose or grade later to preserve the intended look.
• Raise ISO and use the fastest shutter speed you can tolerate. Noise is usually easier to deal with than blur.
• Use a monopod to reduce camera shake while staying mobile.
• Image stabilization helps camera movement, but not subject movement, so prioritize shutter speed when possible.
• Flash or added constant light usually won’t help in fog; it can make the haze more visible.
• Lean into the look: silhouettes with strong backlight can work very well in haze.
• In post, modest contrast/black level adjustments, or even desaturation/black-and-white, can help cut through the fog and make noise look more intentional.

In short: improve the lighting if you can, stabilize yourself, accept higher ISO, and work creatively with the haze rather than trying to eliminate it completely.