"Hard to turn over" is a tough description.
The starter motor and battery are what "crank" the engine. There is enough energy in a good battery to power the starter motor and turn the engine for several minutes, even if it's out of fuel, has no spark plugs, etc. Just because you CAN, DO NOT turn for several continuous minutes; you can overheat and ruin the starter motor. The owner's manual should have a similar warning.
Colder weather means thicker oil, which mean more power is required to turn the engine. It will turn slower, but should still go a long time.
If it turns slower than usual, and it's not unreasonably cold, the entire system of battery, battery leads, and starter motor should all be inspected. A good battery can't push enough energy down corroded wires or through corroded connections. A starter motor with worn brushes isn't as efficient and doesn't make enough power.
Very large engines sometimes have a starter engine which runs on the same fuel as the main engine. Imagine using your car's engine to start a REALLY BIG engine.
Long story short, the function of the battery and starter motor is to turn the engine, gas or diesel, fast enough until it is capable of running on its own.
Now, if it turns fast enough but doesn't run on its own after a reasonable time, you need to find out why. Each cylinder needs air, fuel and either a spark or compression ignition source. Air, fuel quantity and ignition energy are basic. Spark ignition timing is critical for a gasoline engine. Too soon and the cylinder tries to turn the engine backwards. Too late and you don't make power, you make a flamethrower with burning gas flowing out the exhaust valve but not doing anything really useful.
The occasional puff of black smoke tells me that some fuel injected into the engine isn't burning properly and went into the exhaust system as soot.
It can be too much fuel and a weak spark. It can be the right amount of fuel and a good spark at the wrong time. It can be 6 cylinders working well and 2 that don't. The list goes on and on.
Once you get it started, take it to a Tech who has and uses good diagnostic equipment. Like a good doctor uses an ECG to check your heart, a good Tech can tell you some amazing things about your engine.
Best wishes for a prosperous new year