'I would rather find him private secretary to the first Lord than anything else.'

Admiral Crawford's influence, and Sir Thomas's lack of influence are a bit of a puzzle to me. I have not really solved it, but here is what I have found...

The Naval historian Brian Lavery illustrates the comparative ease with which Frank Austen was made (due to the influence of Admiral Gambier) with the less certain path of another officer in 1800:Many officers had very little influence outside their own ship, and had to follow a very slow and uncertain route by becoming first lieutenant of a ship though seniority, and then being promoted when the ship distinguished herself in battle. This was a policy which had been introduced by Lord Anson in the 1750's . It had both advantages and disadvantages; in particular, there was no guarantee that a ship would find an opportunity for such distinction.
There seems to have been a group of highly competent and experienced senior lieutenants who were much sought after as first lieutenants, but had little immediate hope of promotion. One such was William Pryce Cumby, from a family which was struggling hard to climb the naval ladder. His grandfather had been a master, his father a captain and two of his mother's brothers were lieutenants. Born in Dover, he entered the service at the age of 13. In the Mediterranean Fleet he was known as a 'Merry, clever little fellow' and an 'excellent officer'. He produced a Biblical satire on St Vincent's style of leadership.
Copies were circulated until it came to his lordship's notice. He forced Cumby to read it out to the assembled captains, much to his horror and their amusement. Then St Vincent, who was not lacking in a rather cruel humour, gravely pronounced a verdict:
Lieutenant Cumby, you are unanimously found guilty,and without a court martial, Sir,of parodying the Holy Writ, and that too for the purpose of bringing your commander-in-chief into ridicule... The sentence, therefore, upon you without a trial, I do adjudge,-that for this your offence you have my permission of three months leave of absence in England, to, I hope, amuse yourself there, as you have amused me here; and that, on the day in which you report yourself returned, you do me the honour of dining with me. (1)

William Price-Cumby's horror at being thus singled out is perhaps better explained by the first Lord's use of the unifying force of religion to stifle mutinous behavior:
"In an incident of July 1797 Jervis [John Jervis, later Lord St. Vincent] ordered that the sentence of death, passed on four men by court martial, should be carried out the next day, even though it was Sunday. In the case of two of them, facing charges of mutinous behavior, some of the most incriminating evidence was obtained through confession to a chaplain, normally sacrosanct...he had the ships' companies at prayer with the corpses left swinging at the yardarm...The stern formality and pomp of Sunday worship under st Vincent's flag gave young Lieutenant Pryce Cumby material for his humour. In a parody of the biblical story of Nebuchadnezzar, Cumby cast his admiral as the Babylonian potentate: having set up an image of blue and gold he required all to bow their heads when he appeared for Sunday divisions, and he had his own peculiarly naval punishment ready for anyone failing to strike headgear at the appropriate moment.(2)

Still, neither William Price nor William Pryce Cumby proved deficient in useful contacts- they both were made Lieutenants about as quickly as could be. William Pryce Cumby found a patron in Lord Mulgrave, a politician who was in the government for all but one year of the war, and was First Lord of the Admiralty from 1807-1810. Cumby was a Lieutenant for several years, but even some Captains (for example, Francis Austen) never got command of a frigate, and Cumby commanded several (acting as master of the vessel on prizes, and acting Commander of a ship-of-the-line) before he made Post. 1805 found him first lieutenant of the Bellerophon, a 74 gun ship-of-the-line, and at Trafalgar, where his rise to glory was every bit as steep and as heroic as anything William Price could imagine: his gallant captain having fallen early in the action, [and the master surviving only long enough to tell Cumby of his good fortune]he succeeded to the command of the ship. The Bellerophon's crew engaged, boarded, and captured two line-of-battle ships simultaneously...Lieutenant Cumby was ,in January 1806 promoted to the rank of post captain, passing over the intermediate step of commander. ..appointed in the same year to the command of the Dryad Frigate. He afterwards became flag captain to Vice-admiral B.S. Rowley, and commanded the Hyperion"(3)

Although John Jarvis had been a successful Admiral in active service, neither the first Lord nor his secretary required naval experience. Lord St Vincent wanted a professional service, without corruption and mismanagement, but he was replaced as First Lord by Lord Melville, who was under investigation by one of the anti-corruption commissions he had put in place, for suspicious use of funds while acting as Treasurer of the Navy. St Vincent loathed patronage. On his leaving office "When the King asked if the navy were a better institution now than it had been on St Vincent's entrance into it. St Vincent replied that it was not. He stated: "Sire I have always thought that a sprinkling of nobility was very desirable in the Navy, as it gives some sort of consequence to the service; but at present the Navy is so overrun by the younger branches of nobility, and the sons of Members of Parliament and they so swallow up all the patronage and so choke the channel to promotion, that the son of an old Officer, however meritorious both their services may have been, has little or no chance of getting on."(4)

Still, William Price wishes the acquaintance of the first Lord's secretary, as most officers did, regardless of their candid opinion of the competence of the encumbent. In St. Vincent’s time, this was Evan Nepean, his old purser and the Member for Queenborough. In 1807-09 Arthur Wellesley's older brother (MP for Queens County) held this office, and after 1809, John Croaker, anti-corruption campaigner, friend of Wellesley, Tory Satirist and Critic. That Sir Thomas has no influence at all makes me wonder if he might be a member of the 'Country Party' (not a real political party, just what squires who owed no party their seats called themselves) rather than a Tory. Or maybe this part of the novel was written earlier, or on the assumption that Whigs were as likely as not to join the ministry, as they had done in 1801 and 1806 (assuming Admiral Crawford was a Whig - I think it would be resonable to assume at least that his political allegiances were the opposite of Sir Thomas's). It is a sign of the times that William speaks as if the incumbent could be any MP - after 1812, neither the First Lord or his secretary changed for 15 years.

References:
(1) Brian Lavery, Jack Aubrey Commands, Ch3 p69 (Conway Maritime Press:London,2005)
(2) Richard Blake,`Evangelicals in the Royal Navy, 1775-1814,Ch3 p86 (Boydell Press:Suffolk,2008)
(3) William Fordyce,The history and antiquities of the county palatine of Durham,Vol1 Section 25 p529 (A. Fullarton and co.:Newcastle, England, 1857)