Howdy
Most guns come from the factory with hammer springs that are strong enough to be used as Mack truck suspension springs. They are deliberately made strong enough to reliably fire any brand of primer. It is very unusual to come across a gun that is still in factory condition and will not reliably fire any brand of primer. You do not mention if your Henry has been slicked up at all or not. Once you start lightening hammer springs for faster actions, then you are starting down the road to possible FTFs. You should be setting off factory ammo with 100% reliability if the gun is factory spec.
Since the prolbem is more severe with your reloads, I would look at the reloads themselves as the prime suspect. Are your primers fully seated? A high primer can seat itself fully as the firing pin first strikes it, absorbing some of the force of the blow. In marginal cases, this can be enough to lead to FTFs. Be sure your primers are fully seated. They should be flush with, or a few thousandths sub flush of the case head.
The next suspect to reliability is friction in the action, or anything that prevents the firing pin from doing its job efficiently. Remove your firing pin, and be sure there is no built up crud or fouling in its passage way through the bolt. You might want to lightly polish the surface of the firing pin while you are at it, to slick it up a bit. Just a very light polishing. Frankly, I do not recommend messing with the firing pin return spring. You need it to continue doing its job of retracting the firing pin, even though the firing pin has to overcome the spring on its way to the primer. Be sure the hammer spring is snugged down tight so the spring can deliver its maximum effort.
That's really about it. There are a couple of tricks you can do to the firing pin, but now we're starting to talk about a 'race gun', and I really don't want to go there.
When I had Happy Trails do a complete action job on my Henry, he lightened the hammer spring, as well as performing a bunch of other tricks. The Henry does not come with a strain screw for the hammer spring, like there is on a '73. I had Hap install a strain screw on my Henry, so that in case the hammer spring was a tad too light I could tweak it. I've only used my Henry in 3 matches so far since the action job, and on Sunday I had a couple of Failures to Fire. So I need to turn in my strain screw 1/8 turn or so to increase my reliability. But my Henry is pushing the envelope of reliabilty with its light springs. On a stock Henry this should not be a problem.
I can't imagine a stock Henry with a stock hammer spring giving failures to fire. It might be traceable to your primers, and as was said, be sure your hammer spring is snugged down tight.
Other than that, Federal primers may be the answer. They are easier to set of than Winchester. I have a Colt with a very light hammer spring and it gets a little bit fussy about Winchester primers, but it always sets off Federals. But a factory stock gun should be able to light off commercial ammo 100% of the time.