Why do unsaturated hydrocarbons burn with a sooty flame while saturated hydrocarbons burn with a clear flame when burnt in the air?
Your observation that unsaturated hydrocarbons burn with a sooty flame is not always true and depends on the circumstances. There are more factors at play than the nature of the hydrocarbon that is burning.

The most important reason why flames are sometimes smoky is that the combustion is incomplete. This means that not enough oxygen was present to fully oxidise all the carbon to carbon dioxide or carbon monoxide gases (and the amount of CO will vary with the ratio as well as, given enough oxygen, it should also be completely combusted to carbon dioxide). If there is not enough oxygen in the flame there are many competing side reactions that generate sooty resides (the reactions are pretty complex). Some pyrolysis reactions of hydrocarbons may occur before combustion itself (if the mixing in the flame is less than perfect) and these may contribute to the generation of sooty products which may be harder to burn than the original hydrocarbon.

Incomplete combustion is as much a function of the environment of burning as it is the thing that burns. Some environments don't enable enough oxygen to get into the gas mixture to drive complete combustion and the reactions can easily leave a lot of sooty emissions. A candle, for example, burns with a smoky flame despite being made from mostly saturated long chain hydrocarbons. It does so because a candle wick doesn't do a good job of mixing air with the vapours of the burning hydrocarbon (this is deliberate as the prime role of candles is to provide light and this comes from incandescent particles of soot in the flame: a pure flame burning efficiently would provide little light by itself).

Even well controlled environment such as engines can't completely avoid side reactions. But Diesel engines (which use mostly saturated hydrocarbons) produce more sooty particles than petrol (gasoline) engines though petrol engines have far more unsaturated fuels. The exact reasons are complex and depend on the precise environment inside the engine.

In well controlled environment where the oxygen-fuel ration is set carefully most hydrocarbons burn cleanly with little soot. Gas-based cookers (using mostly saturated methane, ethane, propane or butane) are very clean because they control the gas mix very precisely (use the wrong burner and they may well be sooty!). Welding and cutting torches can happily use the highly unsaturated acetylene with a clean, non-sooty, flame (the oxygen-acetylene flame is one of the hottest achievable) because they control the gas mix very carefully leaving little room for side reactions.

Uncontrolled flames might well show your general observation that unsaturated fuels burn more sootily, but this is as much a product of the environment as it is the product of the hydrocarbon. It is hard, without a carefully controlled experiment, to tell whether the general observation is true at all.