There is a lot of mystery with this story, like why do they want to replace the short block because of leaking valves. There are many unknowns with this problem.
Customer understanding and technicians communication skills are a common problem. If doctors had the same communication skills as mechanics we would fear dying after each exam.
Thanks again for all the info and assistance. I have attached several more pictures that were taken by my insurance company . Once I hear back from Hyundai, I plan to meet with the owner of the dealer and then insurance company wants to send someone else out to inspect the motor and my thoughts are to maybe have them move it to another dealer for repairs and get their opinion on what caused this .
@Bobbyg1958
What Is Actually Damaged? Why Canāt It Go Back Together With A New Head Gasket And Bolts?
CSA
ok4450: āI could be wrong but that wood also looks to be something soft such as pine and not oak. The end of a shop broom comes to mindā¦ā My best guess would be a handful of wood chips from the dealershipās landscaping beds.
The debris wedged in the intake valve and the smaller piece on the exhaust could certainly explain the 25 PSI of compression.
Seeing as how this is about as strange as it gets I might pose this scenario. Assume for the sake of discussion the OP drove the car for 13 miles with no issue until BAM! doom hits.
The OP has essentially a new car which I assume had been sitting on the dealer lot for quite a while seeing as how itās a 2014. In truth it should have been sold already. Could it be that rats gnawed their way into the intake tract and started setting up house while it was on the dealer lot?
At some point packed debris could have been shaken and sucked loose and which has now led us to this point?
IF that is the case then it would be a good idea to open the air filter housing up to see what, if anything, is in there. Just something for consideration since ānormalā stuff is off the table. 
It appears the wood entered through the intake manifold. When was the last time the air cleaner was opened for inspection? Did you just get an oil change?
I like OK and Nevadaās comments. I must have missed the part about it being a brand new 2014? Regardless of where it may have happened, if it is rodents, is that an insurance or dealer warranty issue? I dunno, donāt think so. Vandalism by mouse? Yeah you would have thought theyād inspect the air filter and intake with gloves on, just in case.
I do remember way back coming home after a two week vacation and finding a mouse nest in the air cleaner of the car that was just sitting in the garage for a couple weeks. It was the old style air cleaner with the carb and the house was only a few years old so not really something you would expect. Except there was a corn field adjacent to our back yard that seemed to encourage rodent activity. Those pesky devils. Cats are manās best friend.
āCats are manās best friend.ā
Snakes are also very effective at controlling rodent infestations!
Unfortunately, many people become upset at the sight of snakes.
I just pick them up and toss them over the fence onto the property of the farm in back of my house, but Iām pretty confident that they help to keep vole (āfield mouseā) problems under control on my property.
Lets see, vole or snake, vole or snake, its a toss up and I hate that vole or two, but I think I dislike snakes more. I still remember my mom running screaming down the driveway after seeing a garter snake.
I knew I had seen those wood chips used in landscaping. Photo of them in neighbors front yard with possible 99.9% DNA match.
I am going to vandalize someoneās vehicle. 1. manage to open hood without leaving signs of doing so 2. Have spark plug sockets with me to fit any size plug. 3. Push mulching wood chips in hole and replace plug That is to much effort so I will think of something else.
Maybe I should retract my accusation of someone sabotaging, or rather staging, this problem with wood chips.
Iām leaning more towards rat storage since this car may have languished a long time on the lot.
I never felt from the start that someone would be bored enough or vindictive enough to remove a spark plug for vandalism purposes.
@ācommon sense answerā writes ā¦
GM cars there's a fine mesh "screen" .... in the opening of the throttle body intake throat.
Thatās actually a pretty good idea. My 4afe equipped Corolla doesnāt have that feature though.
The last of the GM carbureted and early TBI models also had screens. One very cold night on the way home I came upon a lady broke down in her Blazer or whatever it was. She said the engine was screaming wide open all the time and the throttle body had been worked on recently.
She had a flashlight on her (I was on a motorcycle) so I removed the air cleaner and crawled up there to look around. I found a throttle body part lying on a screen and which had gotten caught up in part of the linkage.
I had no tools but luckily she had some chewing gum. I found a stick on the roadside, stuck the gum to it, and fished the part out with a little wrestling. All was well then and she motored on home.
When I reconfigured my induction system on my Scion I discovered a carbon āfilterā permanently mounted into the top of the air filter housing. It isnāt shown on any parts drawings or on the parts list. I finally dug deep enough and discovered that itās there to prevent any hydrocarbon molecules from finding their way past an open valve and out to freedom after the engine is shut off. I removed it.
Having seen the bits of wood stuck in the valves in the new photo group, I can accept the possibility of a rodent somewhere in the induction system as a source. That would also explain why it didnāt manifest itself as a problem until the car had been driven some. It would have had to work its way through the induction system. I still find it awfully weird, but Iāll just have to add it to my list of lifeās weird things.
I doubt if either the warranty or the insurance will cover it. But that shouldnāt discourage the OP from trying. To the OP: sincere best.
IF this problem is indeed the work of rodents (should be verifiable by inspection of the intake tract) and If the dealer and Hyundai have one iota of honor they should install a new engine in this car or preferably just put them behind the wheel of another new car.
Granted, itās not a factory defect but itās certainly not due to anything the OP did. Push corporate Hyundai to Good Will warranty the thing.
I wonder if thereās any mulch around shrubs or walkways at the dealer that is similar to whatās in that engine? If so, maybe picking up and preserving a handful would be a good idea along with mailing a sample to corporate Hyundai.
Maybe thereās something around the dealer that resembles the mulch below in the first pic.
I found an example of the landscape wood chips in a neighbors yard which were identical to the OPās photos. I photographed them but although following directions was unable to post my photos. Some had discoloration due to darker spots in the wood.
Iāve got about 50 bags of it around the evergreens. Come get a handfull.
The OP did not state when the car was purchased, could have been in 2013. This is not a new car, looks like at least 20,000 miles on the pistons and valves.
Nevada ha s a very good point. Iām thinking that I misread the original post and interpreted it as an aged new car only having 13 miles on the clock when it actually meant they had only driven the car 13 miles that particular day.
My bad, my error, oops, kick self. Now all bets are off. 
So now that gets into the point of whether the OP has mulch like that at their house or at a next door neighbor.