Does this deal an extra 2d6 radiant damage with every weapon hit for the full minute? Or is it lost once the weapon hits once?

"The NEXT time". Unfortunately only procs on your first attack.

Yeah researched and agreed. All of the Paladin "smite" spells only deal the bonus damage on one hit.

I am trying to add this spell to my 8th level Eldritch Knight, which should be possible because Eldritch Knights focus their study of magic on the abjuration and evocation schools. Further, page 75 of the PHB says, "The spells you learn at 8th, 14th, and 20th level can come from any school of magic."

However, this spell is not shown as an option when I go to add it from the Manage Spells u.i. I had to create a Paladin (and then level him up to 5th) before the Branding Smite spell would appear as an option to learn (or, in the case of the Paladin, "prepare").

Is this simply a bug? I have confirmed that I have the available spell lots of the proper level, and many other spells were available to be learned/added, just not Branding Smite.

Eldritch Knight can only learn wizard spells.

why does it say "hit a creature with a weapon attack" instead of how other smite spells are worded as- "hit a creature with a melee weapon attack"

i know its probably so you can used a ranged weapon for the spell but it also prevents the use of triggering it with an unarmed strike as i have found out in the monk section of sage advice compendium that either unarmed strikes count as melee weapon attacks for triggers always or you have to multiclass into monk to be able to use spell triggers or other things that activate with the wording "melee weapon attack". Just as stunning strike is triggered with a melee weapon attack and it can be used with an a unarmed strike

https://www.dndbeyond.com/sources/sac/sage-advice-compendium#ClassFeatures

I think the intention is solely to allow both ranged and melee weapon attacks. An unarmed strike would still count as a weapon attack for the purposes of this spell, because there are only weapon attacks and spell attacks. A dragon's claw attacks still count as weapon attacks, despite not using a weapon, for example.

Now to make a dragon of smite 

branding smite

Inst this just a terrible version of the paladins own Divine Smite class feature? (unless your fighting invisible enemies)
I guess you can still stack them like cast this as a bonus action, take the attack action, and during the attack still add divine smite on top of it if you really want to burn through your spellslots fast

Needs to be available for E. Knight

No advantage on the next attack(s) of still concentrating?... seems exactly like Fairy Fire

one of the Tiefling variant from Volo's get is for free

Quote from AkumaHellkite >>

No advantage on the next attack(s) of still concentrating?... seems exactly like Fairy Fire

 Yup, the difference between this and Faerie Fire is that there's no save attached to this spell and it deals damage as well. So it's a nice way to guarantee that whatever you hit won't be going invisible while you're concentrating on the spell as it goes right through lucky saves and legendary resistances.
It's differently useful, and not always better in every situation, as no spell should be the cure-all for every problem lol

It's lost after the first hit.

Inst this just a terrible version of the paladins own Divine Smite class feature? (unless your fighting invisible enemies)
I guess you can still stack them like cast this as a bonus action, take the attack action, and during the attack still add divine smite on top of it if you really want to burn through your spellslots fast

I mean yeah, that's what these smite spells are for. They're there to provide even more damage and add utility to your Paladin's smiting. Also so that subclasses like the Hexblade and Battlesmith can have some form of smiting (ik Eldritch Smite exists too).

Quote from Mr_Pockets >>

why does it say "hit a creature with a weapon attack" instead of how other smite spells are worded as- "hit a creature with a melee weapon attack"

i know its probably so you can used a ranged weapon for the spell but it also prevents the use of triggering it with an unarmed strike as i have found out in the monk section of sage advice compendium that either unarmed strikes count as melee weapon attacks for triggers always or you have to multiclass into monk to be able to use spell triggers or other things that activate with the wording "melee weapon attack". Just as stunning strike is triggered with a melee weapon attack and it can be used with an a unarmed strike

https://www.dndbeyond.com/sources/sac/sage-advice-compendium#ClassFeatures

 Yeah, it also says "the next time you hit," which means as long as you maintain concentration, even if you miss, the spell can still proc.  This feels like a spell perfect for artificers that use guns. 

Quote from Mr_Pockets >>

why does it say "hit a creature with a weapon attack" instead of how other smite spells are worded as- "hit a creature with a melee weapon attack"

i know its probably so you can used a ranged weapon for the spell but it also prevents the use of triggering it with an unarmed strike as i have found out in the monk section of sage advice compendium that either unarmed strikes count as melee weapon attacks for triggers always or you have to multiclass into monk to be able to use spell triggers or other things that activate with the wording "melee weapon attack". Just as stunning strike is triggered with a melee weapon attack and it can be used with an a unarmed strike

https://www.dndbeyond.com/sources/sac/sage-advice-compendium#ClassFeatures

 Unarmed Strikes are considered weapon attacks.  Also, it is intriguing that this and banishing smite are the ONLY two spells that can be used with ranged attacks.  What's the logic behind that?  These spells were created before Hexblade or Battle Smith, two subclasses where people can focus on either melee or range, and notably, these two spells also exist.  What's the idea of allowing this spell to empower ranged attacks, but not other smite spells like staggering smite or wrathful smite.  

Quote from Fusion2727 >>

Inst this just a terrible version of the paladins own Divine Smite class feature? (unless your fighting invisible enemies)
I guess you can still stack them like cast this as a bonus action, take the attack action, and during the attack still add divine smite on top of it if you really want to burn through your spellslots fast

 

I mean yeah, that's what these smite spells are for. They're there to provide even more damage and add utility to your Paladin's smiting. Also so that subclasses like the Hexblade and Battlesmith can have some form of smiting (ik Eldritch Smite exists too).

 The use of this spell is to reveal invisible creatures.  Otherwise, I'll recommend spending that concentrate slot on haste, heroism, bless, or spirit shroud.  The "smite" spells best saved are for utility, not damage, all because they use up your concentration.  As a Hexblade, concentrate on shadow of moil or the ol' reliable hex.    

Quote from Elunedra >>

Inst this just a terrible version of the paladins own Divine Smite class feature? (unless your fighting invisible enemies)
I guess you can still stack them like cast this as a bonus action, take the attack action, and during the attack still add divine smite on top of it if you really want to burn through your spellslots fast

 It can also work on ranged weapon attacks, which Divine Smite cannot do.