Napoleon was firmly against corruption in his government and the army.

He established the Inspecteurs aux Revues in 1804 to inspect military units' records and accounts. They were responsible for the inspection of regimental organization, administration, pay accounts, property accounts, muster rolls, and personal records. They were hand-picked and had the authority to call out units for inspection, and those units so required had to turn out in full dress. They had an excellent reputation for honesty and finding fraud and false returns from units. During their first year in existence they found 50,000 false returns and those who were responsible were held accountable for their misdeeds with a subsequent damage to their careers. These inspectors were attached to army and corps headquarters as well as the Ministry of War.

For high-level and more complicated frauds, Napoleon established the Auditors of the Council of State who were also hand-picked, and besides their investigative functions, were also trained to be high-level civil servants.

Did the British government have equivalent organizations for the same purpose? Did the governments of Austria, Russia, and Prussia where favoritism and corruption were rampant?

No one is saying something silly about 'goodness and light' except you. There is, unfortunately, always corruption in government and elsewhere. The question is, what is done about it? Seems to me that Napoleon established two organizations to attempt to solve the obvious. What did the British government do to solve their corruption problems? Seems to me that Napoleon and his government were way ahead of the power curve here as elsewhere.