Risk of flat tire in bicycle
Excluding tires so old and worn they obviously need replacement, there are two main root causes of punctures in well maintained bicycles.    

Low Pressure causing 'snake bite' punctures. When the pressure is too low, and you hit a bump, the tire can pinch the tube against the rim and cause a puncture. Small tires have to run at higher pressure to stop this happening and have lower volume. All tires leak air slowly - so the combination of low volume and higher pressure means they  need to be pumped up more often than larger tires - in some cases weekly.
Additionally a slow leak that may go unnoticed for longer in a larger tire will become catastrophic to a low volume tire much quicker. 

Punctures where an object penetrates the tire is what most people think of. While small tires have a smaller area to hit such an object, they run higher pressure and are therefore harder - which generates more force to push the object into the tire. Generally a larger tire spreads the loads over a larger area, so less likely to push an object into the tire. Another aspect is something sharp like glass may cut a tire and the higher pressure of the smaller tire will push the tube though the cut. 

The differences are small, and it depends as much on tire and wheel choice as size, but all else being equal, experience shows that most cyclists puncture less on larger tires.  

As far as punctures go, you need to be able to deal with them on a long ride regardless of the bike you are riding. If the road surface is good quality and generally swept clean, then a road bike is more suited and practically no more likely to puncture than a cross bike. If the road potholed and rough, with lots of stones and litter then larger tires are better choice. Larger tires also offer a more comfortable ride and for this reason may be more suited for you. The rolling and aerodynamic efficiencies of narrow tires may make a difference to a pro rider trying to break away from the peloton, but you may find the few seconds per hour difference in speed less enticing at the end of 200+km or riding. 

Also I suggest you take the bike out for a decent ride before the event. 230km on a bike you have ridden no further than around a car park is a big risk.