Under the face of a padel racket is a block of EVA foam. Its hardness is one of the few specifications that changes the feel of every single shot, and most makers publish it in one word.
Soft
A soft core compresses more easily. The ball stays on the face fractionally longer, which most players read as more control and a softer contact. It suits a slower swing, because it does not need to be driven hard to deform, and it is generally gentler on the arm.
Hard
A hard core compresses less and returns more of the shot instead of absorbing it. That returns pace to a swing that is already fast enough to load it. Applied to a swing that is not, it gives less rather than more, because the core never fully compresses.
Between the two
Makers use several words for the middle: average density, medium, medium hard. They are recorded here as published rather than rounded to hard or soft. A core between the two is the usual choice when a swing is still settling and neither extreme is obviously right.
Compound names are not hardness
Many manufacturers name their foam — you will see trade names in the specification instead of, or alongside, a hardness. A name tells you which compound the maker used; it does not tell you whether that compound is hard or soft. Where a maker publishes only the name, that is what appears on the product page, and no hardness is inferred from it.
Core and frame are different questions
The core is the foam. The frame and the face are carbon or fibreglass, and they change stiffness independently. A soft core in a stiff carbon frame does not feel like a soft core in a fibreglass one, which is why both fields are recorded separately.
Temperature changes the feel
EVA is a foam, and foams stiffen when cold and soften when warm. The same racket plays firmer on a cold indoor morning than it does in summer, and the effect is large enough that players notice it without being told to look for it. A frame that feels right in one season may feel different in another without anything having changed.
The same property is why storage matters. A racket left in a hot car boot is being held at a temperature the core was not designed to sit at, and heat is what ages foam and the adhesives around it. Room temperature and a cover is the whole of the advice.
Core and arm
A soft core absorbs more of the impact and passes less of it back up the handle. That is why soft cores are usually the recommendation for a player returning from an arm problem, and it is worth saying plainly that this is a property of the foam rather than a medical claim. Anyone playing through arm pain should be asking a physiotherapist, not a product page.
Where the hardness is published
Not every maker states one. Where the specification on a product page shows no EVA row, it is because the manufacturer published nothing, not because the racket has no core.