Haze is similar to wash out. Haze from UV must be cleared up at the lens by a UV filter and wash out or overexposing of highlights prevented at the lens with ND filters. Software cannot restore the details that were never recorded.
But modern DSLR lenses already have UV coatings and the camera sensor itself has a UV filter in front of it. So for general use the DSLR already has its UV filter attached all the time and has no need of an additional lens UV filter to cut out UV.
The exceptions are in strong hazy environments (typically most of the way up the side of a mountain

).
I'm not fluent in UV filter and digital sensor but here is an interesting read concerning UV and IR
Another test for UV photography is using a sunscreening lotion to cover a face partially. You should get a clear-cut pattern, because a decent lotion effectively absorbs UV
Nikon D70, UV-Nikkor 105 mm f/4.5,
Hoya-U-360 filter
Photo © Bjørn Rørslett/NN
Why the Hoya Filter? On D70? To maximize UV passage
All You Ever Wanted to Know About Digital UV and IR Photography, But Could Not Afford to Ask
3. Capturing digital UV
All About Digital UV and IR. Page 3: Can Digital Capture UV images?
a tease:
"Exposures for digital UV photography are often handled well by the camera itself, just as we found for digital IR. However, take extreme care to ensure that no stray light can enter through the camera's ocular. Use the eye-piece shutter to guard against this, or if your camera lacks such feature, cover the ocular by your hand or black cloth. Since the UV levels are much lower than ordinary light, the metering system is guaranteed to be thrown out of whack unless you take these precautions.
Some cameras, in particular Nikon D100, needs a substantial +EV correction to be dialed in as well. Similar to the IR case, you should check that the LCD histogram displays really useful information. If the CCD records little in the green channel for the UV pictures, the histogram will indicate a severe underexposure, while in fact the image may be properly exposed for UV. You have to examine the entire RGB-histogram in a suitable computer program (Bibble, Nikon Capture, Adobe Photoshop etc.) to make sure your exposures are spot on.
Compared with visual-light shooting, you can expect any kind of UV photography to give from 6 up to 15 stops longer exposures, so by necessity a tripod should be used. The apparent loss of film/digital sensitivity is largely caused by UV intensity itself being so much lower than visible light levels, but a further loss occur due to lens and the recording media (film or digital sensors). With my UV-Nikkor lens, I find the digital UV correction to range from +4 to +6.5 EV, which in fact is quite similar to that of film-based UV photography. However, for non-quartz lenses you would expect quite higher correction factors than this. Stacking the UV bandpass filter(s) with a hot-mirror filter also gives a penalty measured in a higher factor, thus typically 1- 2 EV should be added.
One might end up with excessively long exposures, so digital noise is likely degrading the images. If your camera has a noise-reduction feature, ensure it is switched on before shooting UV.