So, Maneuvers in Tome of Battle come in three shapes (strikes, boosts and counters), which make the martial characters extremely deadly, even through they are usually limited to 1 boosted attack per round... Not that they usually need more than that. But what if they do? The feats "Cleave" and "Great Cleave" allow one to strike another foe if you've killed the previous one, using the same modifiers and such, even if your BAB does not allow you to hit multiple times. Do these feats synergize with martial maneuvers? Let's say, I boost my attack with Burning Blade and use Mountain Hammer, and drop my enemy. Does my Cleave attack use Mountain Hammer and Burning Blade as well?
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Each Maneuver Must Be Evaluated Individually...
There's no way to generalize the applicability of every maneuver to every ability that grants additional attacks. Some can be used and some can't.
In your example, while the 1st-level Desert Wind martial maneuver burning blade (boost) (ToB 52) does last, like its duration says, until "End of turn," but the 2nd-level Stone Dragon martial maneuver mountain hammer (strike) (ToB 84) says, "As part of this maneuver, you make a single melee attack." That really means only one single attack gets the benefits of the martial maneuver mountain hammer, even if the target's dropped by a creature using the martial maneuver mountain hammer who has the feat Cleave (PH 92). That's because buried in the Blade Magic chapter in the section (Type) under the heading Strike there's this text:
Thus an adept possessing the feat Cleave who initiates the martial maneuver mountain hammer who drops a foe does get to make the attack granted via the feat Cleave, but he gains no benefit from the martial maneuver mountain hammer when he does. To be clear, when he's granted that extra attack from the feat Cleave, the adept isn't benefiting from "extra attacks when making a strike" but benefiting from extra attacks after resolving the strike--that is, as a result of the strike.
...But Many Boosts Last Until the Turn Ends
In the heading labeled (Type) the section Boost reads, in part,
Emphasis mine. Therefore the martial maneuver burning blade lasts until the end of the initiator's turn because it affects attacks, but the duration of 6th-level Diamond Mind martial maneuver moment of alacrity (boost) (ToB 64) remains instantaneous.
Initiation Time: One Standard Action
You can use a standard action to attack, but you can't use an attack (sometimes called an attack action) to perform a standard action. You're essentially 'trading down' when you decide to use a standard action to make an attack.
Initiating Mountain Hammer is a Standard Action, not an Attack, so the Attack/Attack Action granted by Cleave or Great Cleave is not eligible to be used to Initiate a maneuver.
Mountain Hammer only makes one attack (with bonuses), it doesn't affect anything else that turn. A few Strike maneuvers do give bonuses that can apply to AoOs or extra attacks, but they specifically note that they do so. Boosts, on the other hand, if they last for an entire turn, explicitly apply to any attacks you make that turn.
So, in your example, Burning Blade would apply to the Cleave attack, but the bonuses from Mountain Hammer would not.
There are no Strike maneuvers in Tome of Battle: Book of Nine Swords that may be initiated as an Attack action - however, I wish there were, as it would be thematically appropriate and provide a larger array of options for Sublime characters.