When my son was 9 I took him hunting with me for squirrel, and when he showed me he could sit still and quiet I told him I would take him deer hunting with me that year IF he could keep slugs inside of a pie plate at 25 yds...
I bought him a used H&R 20 guage modified just like you have, only older before H&R became New England, and cut the stock down with a hacksaw in 2- 1" increments after drilling 1/2" holes horizontally through it, mounted a "slip on" butt pad with duct tape, so the gun fit him, and took him to the range for two weekends with Remington "Sluggers"
I remember him running into the house to show Mom his paper plate with 5 slug holes in it.
I built him an 8 foot stand out of 2x4s, put it up with him on a parcel of state land only we knew about and I had killed a few deer on (THEN, now it seems a HUNDRED other hunters have discovered it...)
And two days before the season started I fell off a roof of one of our stores investigating a burglary and ended up in the Hospital....
I got discharged, and hunted on crutches painted in camo just so he could hunt with me!
I had "blazed" trees around his stand with a 'hawk at 25 yds and told him he could shoot if a deer was within the marks...
He saw deer, but NOT within the marks so didn't shoot...
The next year I had permission to hunt private land, and he saw 27 deer the whole season...but NONE within the range I gave him, so no shots.
The NEXT year I put him in his stand, and hunt from mine about 200 yds away...opening day, and I hear a shot....then another, then his whistle, which I gave him to let me know to come now....
I spent a couple of hours of PRIME opening day "Magic Time" tracking his doe, which "Jumped straight up like Grandpa said they ALWAYS do if you hit them in the heart...." (I found the tracks, AND the big gouge in the mud where the slug probably went under her...she JUMPED from the MUD hitting her belly ,LOL....)
In fact, he missed first, and when reloading he dropped the slug which clattered down the steps of the stand and the deer just STOOD there, and his THIRD slug is when he "Swore he hit her!" But no blood, no trail....
The NEXT year my stand was only about 150 yds from his, and I watched the young 6 or 8 pt buck be-bopping up the fence line RIGHT towards his stand, with my front sight on him....and when he was about 20 yds away from my son the buck suddenly turns and bolts back the way he came!
I fired, it was about 100 yds...I think I grazed him because he spun around and ran RIGHT at me...5 shots over his back, he STOPS right under my stand at about 10 yds...and "CLICK"... I am empty....and he bolts away...
We BOTH learned a lesson that day...he, to COCK the darn shotgun when you FIRST see the deer a hundred yards away, NOT when he is UNDER your stand....and ME, aim at his FEET when he is running AT you when you are in a tree stand....(I have missed a LOT of deer like this btw....

)"leading" a deer running AT you is hard to do...you have to have your front sight in the GROUND in front of him!!!....
Anyway, we did not find it or a blood trail, that winter I bought him a .50 Cal CVA Frontier Carbine kit which we put together and he shot all summer and then killed his first buck the next year at 40 yds moving, when he was 13...
BUT that shotgun "grew with him," 1/2 dowel rods in the holes I had drilled in the butt allowed me to put it together in increments so it continued to fit him, and he killed a fair amount of squirrels with it too...
I still have that shotgun, put together all the way with the original buttplate on it...
His daughter is 18 months old...and I am already about to take the sections off the stock so it will fit her...in a couple of years SHE will hunt with me with that shotgun....
Long story short....YES....slugs CAN be shot through ANY shotgun barrel...
However, smoothbore barrels with choke are DESIGNED to shoot SHOT....
SOME barrels, even with extra Full choke will shoot slugs WELL....but some, even cylinder bore barrels DESIGNED for slugs will NOT....
And two IDENTICAL shotguns made one serial number apart will shoot slugs very differently....
Bottom line, ANY smoothbore shotgun barrel of any length with any choke CAN handle slugs. You have to fire many differerent types of slugs through it to find the type it likes, and if it is smoothbore, do NOT waste your money on Brennekes, or any expensive Sabots, etc. Stick with standard Foster type slugs, and try Winchesters, Remington Sluggers, or Federal "Trueballs...."
If you get a decent group, KEEP it. IF not, SELL it and try another, even of the same type!
No matter WHAT make or model, you get a smoothbore that will keep standard Foster slugs inside 4-5 " at 50 yds with just the bead on the end of the barrel, KEEP it. All you need to fiddle with is SIGHTS.
You get one that will keep them within 2" at 50, LOCK THAT ONE UP and decide now who will inherit it when you die....
That type of 20 guage you have has killed PLENTY of deer...jut remember...25-50 yds is optimum, but with a 20, your MAXIMUM range is about 80-90 yds....
With a 16, your MAXIMUM range is about 90-100...
With a 12, your MAXIMUM is 100-110...PERIOD.
ALL slugs any guage start with about the same muzzle velocity (fps) Even the .410 slug will do a good job on deer out to about 50-60 yds....
Slugs do NOT fly like rifle rounds, but like badminton "shuttlecocks" (Weight forward)
Range is STRICKLY dependent on the WEIGHT of the "ejecta," or the slug....which is "retained energy...."
Now YEAH I killed my monster buck at 192 yds with a 12 guage 1 oz slugger...
BUT I have been shooting my '97 for 20 years with slugs, AND I had a Williams aperture on it with a Williams ramp front on it, AND I knew it would do 5" at 100 with my slugs from a bench, AND the deer was PERFECTLY situated broadside where I could lean forward on my stand like a bench....but I STILL held about a foot over his back when I touched it off...
AGAIN, long story longer

Shoot slugs through that shotgun all you want...it will NOT hurt the gun.
Practical accuracy is limited to how far away you are CONFIDENT you can keep 5 of 5 shots in a paper plate off hand...(the kill zone of a deer is just a LITTLE bigger than a paper plate.... about 10"x10")
If it is 25 yds like my 9 year old, so be it....keep your shots within 25 yds....but a NORMAL person offhand should be able with practice to keep ANY shotgun within 8" or so out to 50 yds....and since MOST deer (at least EAST of the Mississippi, are shot at 50 yds or less so you are SET