Haze in an older lens is usually caused by lubricants used on the lens slowly condensing over time inside the lens. Fungus can also cause a hazy look to a lens if it grows evenly in the lens, although most often fungus looks like radiating lines or spots within the lens.
How much does haze really effect your pictures? Here's an example of a 1935 Leica Summar 5cm f/2 lens. This had some significant haze that was cutting down on overall contrast and also increasing flare in the lens. I removed the lens elements and cleaned it with alcohol which removed a lot of the haze. It still had a good amount of "cleaning marks" (small scratches caused by cleaning the older, softer glass over the years).
Here are before and after pictures of the same subjects with similar lighting taken on a Fuji X-E2 with Leica screw mount adapter.
These are a bit exaggerated because in both of them, the morning sun is shining from slightly to the right of and slightly above the frame.
Here's another example with no sun flare.
Exposure isn't exactly the same on these last two pics, as they were taken on different days, but you'll notice more detail and sharpness in the after picture if you look at a large version.