I have tried to use the new arc warlock melee aspect with the charged with light mod called "Heavy Handed". On use, it should consume one stack of charged with light to restore half my melee energy, but it doesn't even consume the stack on use. I have tested this multiple times and only managed to get half of my melee charge back once, so I presume that Heavy Handed should work with the new arc warlock aspect melee.
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Lightning Surge, Consecration, Howling Storm, and Tempest Strike are considered "Aspects" or "melee modifiers" not directly considered a melee ability. So anything that says "using your melee ability" refers to the charged melee ability you selected in your Subclass tree, not the Aspect. The only thing that these melee modifiers benefit from are exotics and buffs that increase "melee damage" such as Wormgod Caress, Liar's Handshake, and Winter's Guile, or perks like One-Two Punch. Tempest Strike can be buffed by Cross Counter, but not by Combination Blow, which only buffs Combination Blow itself. Now here's the cool trick: any additional effects triggered by a melee (Scorch, Ignite, Jolt, etc) will ALSO benefit from any buffs that effect the "melee ability" so warlocks can hit 400,000 dmg Ingites and 100,000 dmg Jolts, and Titans can hit 700,000 dmg Ignites with the right setups. Hunters can make dozens of orbs for teammates using Combination Blow, Flow State, Lethal Current, and Striking Light (CWL mod). Because not only the melee kill but any enemies killed by the Jolt will count as a melee ability kill and generate an orb.