There is not an official record of that. The damage wouldn't necessarily be considered against the "English King" but against citizens. Enter into that the Spanish, French, Dutch, and other Europeans became victims of the pirates. Overall, it's very hard to calculate the level of damage done b y them, other than "A lot". It would require gathering every single recorded incident, and back then this would be a considerable undertaking. It's still a considerable undertaking to try today, and we have to deal with conflicted descriptions and lost documents if we studied it today. How do you figure the economic damage? What values for cargo do you go with? What if the ship is taken? People who got stolen from or attacked often exaggerate to try and get action taken in their favor. In Dr. Ed Fox's survey of Captain Henry Every's attack on the Mogul vessels in 1695, claims jump between £180,000 and £600,000 in period currency depending on the source of the estimate (and that's just for one voyage).
I do know of one historian attempting to give economic measure to the damage done by pirates during the Golden Age. Dr. Carl Swanson examines the specific case of South Carolina. In his study, he finds that petitions and claims by government authorities in the colonies spoke louder than the actual reports of damages to claim - and that in the 1710s and 1720s, most of pirate attacks took place in a very small window of time. In his article, “‘The Unspeakable Calamity This Poor Province Suffers from Pyrats’: South Carolina and the Golden Age of Piracy.” The Northern Mariner/Le Marin Du Nord 21, no. 2 (2011): 117–142; he concludes in terms of South Carolina and Charleston's losses to the pirates during the Golden Age:
"Pirates seized merchantmen engaged in South Carolina’s transatlantic commerce, closed the port of Charles Town, and even boldly walked the city’s streets with impunity while the governor and Council debated their response to a ransom demand issued by the era’s most notorious pirate captain....Yet piracy did not seriously damage Charles Town’s transatlantic commerce. The number of vessels captured and the duration of piratical depredations were far more limited than contemporary accounts suggest. The pirates also missed the busiest shipping season. In addition, South Carolina’s naval forces hunted down and eliminated pirates, while similar operations mounted by Virginia also enjoyed success. The Yamasee War that preceded the pirates caused greater loss of life and economic damage. Certainly Blackbeard, Stede Bonnet, Charles Vane, Ned Low, and the others were uncommonly bold in their aggression, yet the prominence they achieved was at least partly explained by the interplay of piracy with provincial politics in early eighteenth-century Carolina. Interests opposed to the Lord Proprietors’ government made effective use of the pirates’ actions in calling for the imposition of royal government."
Pirates did do damage to the economy, but was it more serious that the period's wars surrounding it? Evidence doesn't appear to indicate that. People want to think so since pirates are cool and can imagine them ravaging the seas like that.
I think it would be hard to make an actual number for the damage that Pirating brought down. Unfortunately many of my sources are from books so I'll have to source a few key words that I can here.
The Golden Age of Piracy was split up into three separate out bursts and not just one long period. (Although it is attributed as non-stop by some sources.)
The first outbreak was from 1650 to 1680:
This part of the Golden Age of Piracy actually had relatively little to do with the English Crown and focused more on conflicts between French Buccaneers and Spain.
French buccaneers had established themselves on northern Hispaniola as early as 1625, but lived at first mostly as hunters rather than robbers; their transition to full-time piracy was gradual and motivated in part by Spanish efforts to wipe out both the buccaneers and the prey animals on which they depended.
It wasn't until 1655 when the English captured and took control of Jamaica from Spain that we see interaction between the English and Pirates in the "Golden Age of Piracy." And even then it wasn't necessarily all negative. In fact, the English Governors who were in control of Jamaica freely wrote Letters of Marque to both Tortuga Buccaneers and there own countrymen which created a form of "Legal" Pirating.
The continued growth of Port Royal provided these Raiders a place to sell their booty and live comfortably.
In the 1660s, the new French governor of Tortuga, Bertrand d'Ogeron, similarly provided privateering commissions both to his own colonists and to English cutthroats from Port Royal. These conditions brought Caribbean buccaneering to its zenith.
The Second outbreak occurred from 1693 to 1700 and is when we see things get a bit more interesting:
At the end of Britain's Stuart Era when George I ascended to power much of the traditional enmity between Britain and France returned which quickly ended the profitable collaboration between the English Jamaica and the French Tortuga.
The Earthquake of 1692 destroyed much of Port Royal and reduced the attractiveness of the Caribbean to Pirates as their Chief location for moving Plundered goods had disappeared. (This can be read about more in detail in Pirates: An Illustrated History by Nigel Cawthorne)
Many of the Caribbean Governors defended the acting of Pirating against other nations by disregarding Peace Treaties that were signed across the Ocean. Which lead to much of the Spanish Main being simply exhausted; Maracaibo alone had been sacked three times between 1667 and 1678, while Río de la Hacha had been raided five times and Tolú eight.
While this was occurring, many of England's less favored colonies including Bermuda, New York, and Rhode Island, had become cash-starved by the Navigation Acts. This led to many of the Merchants and Governors who were eager for coin and cash flow into their cities overlooking pirating, with one official having been quoted to say "it would be very harsh to hang people that brings in gold to these provinces."
Most of these English borne Pirates targeted Spanish vessels around the Pacific HOWEVER the Indian Ocean provided a much more tempting target as time moved on as the Indian Economic output dwarfed Europe's output at this time due to Silk and Calico exports, which pirates believed to be much more Valuable booty. Additionally there were no real Navies that patrolled the Indian Oceans thus making the shipping vessels far more vulnerable to attacks.
This would set the stage for the famous piracies of Thomas Tew, Henry Every, Robert Culliford and (although his guilt remains a top of debate between historians) William Kidd.
The Final outbreak to occur during the Golden Age of Piracy was following the War of the Spanish Succession and began in 1715 due to the large amount of trained seaman who had been relieved of Military Duty turning to pirating as the sea is what they knew the most and were trained for.
In 1715, pirates launched a major raid on Spanish divers trying to recover gold from a sunken treasure galleon near Florida. The nucleus of the pirate force was a group of English ex-privateers, all of whom would soon be enshrined in infamy: Henry Jennings, Charles Vane, Samuel Bellamy, and Edward England. The attack was successful, but contrary to their expectations, the governor of Jamaica refused to allow Jennings and their cohorts to spend their loot on his island
It was becoming far more obvious that Jamaican Governments didn't want association with Pirates and led to Jennings and Company moving on to Nassau and establishing the Pirate base there. It became their home for 3 years until the arrival of governor Woodes Rogers.
On 5 January 1718, a proclamation was issued announcing clemency for all piratical offenses, provided that those seeking what became known as the "King's Pardon" surrendered not later than 5 September 1718. Colonial governors and deputy governors were authorized to grant the pardon. (This can be read about in detail in Jolly Roger: The Story of the Great Age of Piracy by Peter Pringle)
Pirates began to turn on and target shipping traffic between Africa, the Caribbean, and Europe as it was deemed the most efficient and valuable way of obtaining booty toward the end of the Golden Age of Piracy.
As far as my reading as gone there aren't many records of total amounts looted by Pirates or damages caused but in terms of the "Golden Age of Piracy" many of the pirates that we focus on and immortalize weren't targeting the English but instead the Spanish and then the French after George I's ascension.