With diesel engines white exhaust smoke is an indication of unburnt fuel leaving the exhaust. This can be normal when first starting a diesel engine and will normally go away after the engine has started to warm.
As to the soot, check for an exhaust leak.
If you take it in for service don't be surprised if the tech tells you this is all normal for a diesel generator.
US Army (Ret)
2020 Entegra Accolade 37TS
2019 Jeep Grand Cherokee Trailhawk (Toad)
FMCA - F432054