The description of the Haste spell says, in part (emphasis mine):
Until the spell ends, [the target] 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.
And the Bladesinging wizard's 6th-level Extra Attack feature says (emphasis mine):
Starting at 6th level, you can attack twice, instead of once, whenever you take the Attack action on your turn. Moreover, you can cast one of your cantrips in place of one of those attacks.
As a 6th-level Bladesinging wizard, can you use the extra action granted by the spell Haste to cast a cantrip? What if that cantrip has you make a weapon attack as part of the cantrip, such as Green-Flame Blade or Booming Blade?
Here's how I understand it. You take the Attack action with Haste; then the Bladesinger's Extra Attack triggers when you take the Attack action, which allows you to cast a cantrip in place of one attack. Haste says "(one weapon attack only)" if you choose the Attack action, but it seems like you can satisfy this by making one weapon attack as part of a cantrip, as Green-Flame Blade or Booming Blade specifically have you make a weapon attack.
Haste only seems to care that you:
- Take the Attack action (you do).
- Make one weapon attack only as part of it (you do).
It doesn't have any specific language about the means of how that ends up happening or if a cantrip gets involved in some way.
For reference, the description of the Booming Blade cantrip says (emphasis mine):
You brandish the weapon used in the spell’s casting and make a melee attack with it against one creature within 5 feet of you. On a hit, the target suffers the weapon attack’s normal effects and then becomes sheathed in booming energy until the start of your next turn. If the target willingly moves 5 feet or more before then, the target takes 1d8 thunder damage, and the spell ends.
Likewise, the description of the Green-Flame Blade cantrip says (emphasis mine):
You brandish the weapon used in the spell’s casting and make a melee attack with it against one creature within 5 feet of you. On a hit, the target suffers the weapon attack’s normal effects, and you can cause green fire to leap from the target to a different creature of your choice that you can see within 5 feet of it. The second creature takes fire damage equal to your spellcasting ability modifier.