20HP Single Cylinder Intek won't turn over

Hmmm, doesn't sound good at all.
 
First the camshaft does have ACR and it on the exhaust valve of this engine. Now the 793880 camshaft does have ACR failures so if your not the bump on the exhaust then camshaft would need replacing.
 
Second that pivot stud needs to be Loctite in place and torque to specs.
 
Third with the intake push rod being bent suspect that the valve itself has move and is now preventing the intake spring fully compressing as needed; therefore, bending the push rod.
 
Intake Valve.003-.005 in (.08-.13 mm)Exhaust Valve
.005-.007 in (.13-.18 mm)

I see my thinking is ahead of my typing again sorry. You need to check the intake valve guide since that the side that has the bent push rod. Left the word "guide" out. It is out of place then it is a new head time. These valve guides are non replaceable.
 
As for the ACR is on the exhaust of this cam. And no the engine does not need turn with starter. Once the engine reach near idle speed the ACR swing out and disables itself. Until does reach this speed it will bump the valve.

Well Jacks Small Engine got my order to me just in time.I was able to put things back together yesterday when we had a record 50 degree day,today winter started here,with 0 degree windchill.Some parts were back ordered,they must have shipped order within hours of getting parts and even used USPS priority two day shipping,forcing postal workers to stop reading the newspaper.
I installed new push rod and rod holder,rocker arms and set valve stem gap.Turned over engine by hand,everything seemed to be working,but still no bump opening of exhaust valve.I sprayed a shot of starting fluid in the intake,I have the fuel line disconected and carb bowl empty.Engine fired rite up for a second or two,good news!!I could not help trying to run engine so I installed valve cover and added oil to crank case.I attached fuel line and added fuel to tank.
Turned over engine,everything seemed normal,engine fired rite up and ran great.I guess the decompression valve on the cam is still not working but it is running so good I do not want to take it off mower and take it apart.The only thing I noticed when it is running is a slight chirping sound,could be the intake valve stem was damaged slightly when push rod was bent.I did not mention one of the two valve keeper inserts was missing when I first removed valve cover,could be the cause of the bent pushrod and possible damage to the valve stem.

Thanks for the info Mikel1,it does look like the acr mechanism is on the bottom of the engine,attached to the cam,the intake valve is on the bottom also.And I think you are correct that the cylinder pressure does get released out the intake.I had plugged the intake with steel wool wrapped in plastic to keep water out when I was degreasing the engine,when waiting for parts.I was trying to get it clean before having to split the case.After the parts came and I was cranking engine,after installing rocker arms,new push rod and valve cover gasket.The steel wool and plastic,that I forgot about (Doh!) came shooting out the intake.
After that the engine started rite up and seems to really run good.I did not see much of a bump on the rocker arm for the acr when turning engine over,I could feel something at the push rod just after valve closed.I will listen at the intake when turning engine over for a chirp,or the pop as you described.Hopefully then I would not need to split the case to replace acr,be good to mow this spring,and this post will help others.