So the Haste spell in 5e says that the targeted creature
gains an additional action on each of its turns. That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
This means I technically have a second action, so would that allow me to use the Greenflame Blade cantrip a second time in the same turn (once with the actual action, once with this action) as the cantrip applies to one weapon attack in a similar fashion as a Paladin with Haste could apply Divine Smite onto the Haste-given attack?
No, you can’t cast a spell with your extra action from haste.
Despite including an attack as part of the cantrip, all cantrips use the Cast a Spell action, like any other spell. Haste’s options are limited as you indicated, and Cast a Spell is not among the available options.
But… Divine Smite!
The paladin’s Divine Smite feature is not a spell, despite being fueled by spell slots. Paladins have other kinds of smiting that are spells, but Divine Smite is not. It triggers when you hit a creature with a melee weapon attack, but does not take any sort of action; any melee weapon attack that hits will suffice.