As a Level 6 Bladesinger 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?
You take the Attack action with Haste, then Extra Attack triggers when you take the Attack action which allows you to cast a cantrip in place of one attack. But haste has in parenthesis (one weapon attack only), but you can satisfy this by making one weapon attack as part of the cantrip, as Green-flame Blade or Booming Blade specifically have you make a weapon attack.
Haste only seems to care that you satisfy:
A. Take Attack action (you do).
B. 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.
Further, was the parentheses part of Haste actually intended to limit the attack to "weapon" attacks only, such as preventing a shove or a grapple, or was that just an oversight by the designers?
Choose a willing creature that you can see within range. Until the spell ends, the target's speed is doubled, it gains a +2 bonus to AC, it has advantage on Dexterity saving throws, and it 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.
When the spell ends, the target can't move or take actions until after its next turn, as a wave of lethargy sweeps over it.
Bladsinger's L6 Extra Attack feature:
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.
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.
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.
Emphasis of relevant sentences mine in Bold