In general, running a CoHd20 game shouldn't be very different from running a Dungeons & Dragons 5th Edition game. The basic arrangement of the system is the same. Combat is turn-based. Player characters gain experience from defeating enemies and/or completing quest goals. CoHd20 does bring some specific setting considerations to the table, however, and there may be some surprises in the nooks and crannies. How each of these nuances is address will really come down to the playstyle of each table, but perhaps some guidance can be offered.
City of Heroes provides little detail for how your powers work inside the fiction of the game. You may shoot energy beams from your hands, but the game doesn't care if that power emerges directly from your body, or comes from special magical bracelets, or is the product of high-tech gauntlets. The game can get away with this because it simply avoids any gameplay element where such specifics are relevant. There's no functional difference between them. It's not like someone can, for example, steal those gauntlets.
And if there's a core difference between a CRPG and a TTRPG, that's a good example. Sometimes in City of Heroes, you are defeated and end up not in the Paragon City Hospital but an enemy base. It's more common in City of Villains. But, oddly, you captors seem happy to leave you with all your super-powered gadgetry. In most cases this can be easily abstracted. After all, Energy Blast says nothing about needing certain bits of equipment to make it work. But then you get to a Power Set like Archery, which specifically hinges on you having access to not only a bow, but arrows as well! Convenient that those always seem to be around, isn't it?
Any tabletop GM worth their salt is going to try to strip you of all your weapons, armor, and other items the moment you are unconscious. Certainly you wouldn't be locked in that 5th Column cave cell with your broad sword still on you, right? If the rules had some kind of "can't be separated from weapons" mechanic that didn't offer much justification, the GM would rightly feel overly constrained by that. In City of Heroes, the "GM" is the code running the game, so it's not as bothered. And while some (non-GM) players may scratch their heads and wonder about it, it really just feels like any other CRPG lack of freedom. And hey, it's not a restriction on them, so it's soon forgotten.
This won't do for CoHd20. If you're defeated and captured by an enemy, expect them to separate you from that which makes you a threat if possible. And if that's not possible, there needs to be some degree of credibility around the rationalization why. At the same time, we want to provide this rationalization. "Stealing" powers is a legitimate story element, but a little of that goes a very long way. It should never be routine and really shouldn't be common. So what's a good solution?
Most CoHd20 Power Set features that rely on any amount of components, weaponry, or other equipment specify that at least some of those features will become available again after you complete a short rest. This is an abstracted way to indicate that you have some method of recreating , conjuring, repairing, or otherwise MacGuyvering your gear using whatever materials and resources you have at hand. It also implies using found items as (possibly) temporary replacements. For example, an Assault Rifle Blaster may be able to replicate most of their abilities using a guard's discarded rifle. Maybe not all their abilities right away, but give them a few minutes to work on it in peace and they'll have something adequate.
Another approach, which can work in conjunction with the previous, is that tech-based powers may be designed to work only with you. Perhaps it's keyed to your DNA, or reads your neuron signature, and simply refuses to operate for anyone else. While it doesn't prevent you from being separated from your stuff, it means whoever took it isn't out running around causing mayhem with it. It's likely locked up not too far from where you were, and it's just a matter of time.
Regardless of how this is handled at your table, the notion of powers represented by equipment should be seen as an opportunity to increase immersion and engagement, and not as a way to punish players for their choices. If a player wishes to declare their Archery bow is somehow inseparable from their character, they should be allowed to do so with no more burden than coming up with a loose, general explanation as to why.
Dungeons & Dragons 5th Edition uses the concept of resting as a form of resource management. In fact D&D has used something like this going all the way back. Originally limited to Magic Users and other similar classes, it was described simply that the spellcaster could cast their spells only so many times "per day." As editions progressed, more features that fell outside of magic were tucked into that mechanic, and eventually the ambiguous "per day" description was codified at X number of hours per 24-hour period, doing nothing more strenuous than sitting around, relaxing, and maybe reading. Finally, D&D ended up with the "long rest," which was six hours of light activity and under which the most powerful features (still including spellcasting) were managed, and the "short rest," which was one hour long and could include slightly more vigorous stuff like socializing, repairing gear, and even light travel. Things that still needed some time management but weren't as potent as spells and major features were reset after that shorter time off.
Having two types of rests makes sense in a typical D&D world, where you are often away from the comforts of civilization for days or weeks at a time. In D&D you're usually off in some wilderness or wasteland, exploring long-forgotten ruins, well away from sane folks who don't choose to risk death for a living. Long rests essentially modeled the "roughing it" or "camping out" aspect of such adventures. The short rest was more of a maintenance mode. A way to recharge energies that went toward convenience and casual power. Long rests could make or break you. Put another way, short rests are cheap. Long rests are risky.
This division of rest types makes less sense when you're adventuring within civilization itself. Like everyone, you "long rest" every night when you go home to your apartment. Your individual quests (missions) aren't long enough for it ever to make much sense to take a long rest. Even a short rest is pushing it, as it's not very likely you can spend an hour unnoticed in a Tsoo-infested warehouse.
CoHd20 uses only short rests. While it's possible you could conceivably be trapped in the Sewer Network or Shadow Shard for days, frankly it isn't too likely. The game assumes that your missions last a short number of hours each, and that you can easily grab a snack afterward at a nearby diner. In fact it may even be accurate to view the brief period between ending one mission and starting the next as a short rest, but CoHd20 keeps the specific mechanic just in case you feel the need to camp out in the middle of a mission.
The CoHd20 rules will often say "short rest" when referring to a rest, but this is just to highlight the duration. It helps with clarity. All rests in CoHd20 are short, regardless if they have that label.
Regarding healing and Hit Dice, you can still spend Hit Dice after completing a short rest to recover hit points. You regain all spent Hit Dice and hit points when you actually sleep, which is the closes CoHd20 gets to a long rest. Note that in D&D you only regain up to half your Hit Dice and all of your hit points after a long rest, so even this mechanic is slightly different.