An extensive biography of Robert Heysham is given in this source (http://balder.prohosting.com/shissem/Hissem_Robert_Heysham_Branch.html). The folowing are extracts:

Robert was born in 1758 in Philadelphia. Nothing is known of his youth, but as the son of a successful merchant ship captain in a "cosmopolitan" city like Philadelphia, he would have received a good Grammar School education, at a minimum. There is no indication that he attended college. More likely he was apprenticed like his brother William.
According to Theodore Heysham, a 19th century direct-descendent, Robert was an officer in the American Revolution. A volunteer, he reportedly fought at the battle of Trenton in 1776, when he was 18 years old. Theodore Banta, in "The Sayre Family," also says he "was in the battle of Trenton, when only sixteen [sic] years of age, and continued through the Jersey campaign." Four companies of the Philadelphia militia did take part in the Trenton battle, fighting under General John Cadwallader, though they were not able to cross the river for the first attack because of the ice. The Battle of Trenton was famous, of course, for General Wasington’s dramatic crossing of the Delaware river on Christmas Eve, commemorated in the familiar painting of the subject.

Theodore Heysham, a direct-descendent of the 19th century, claimed Robert was a prisoner of war from 1777-1778. Another reference, which I've suggested refers to Robert's father, William Heysham, simply says that "Heysham" was taken when Howe entered the city. Perhaps Theodore saw this as referring to Robert.
Soon after the British entered and took Philadelphia. As a prisoner of war, Robert would have found himself jailed at the Walnut Street prison. Robert's father, William, had been arrested as a prominent rebel at about this time so the two may have been reunited in jail.

The British retreated from Philadelphia in June 1778 after the French entered the war.
Robert took the oath of allegiance to the patriot cause in October 1779.
In 1783 the war was over and America was independent.
Robert Heysham probably became a Captain of the militia at this time, apparently in the 3rd regiment.

Robert soon got a federal job working for Robert Morris, at left, in the Office of Finance. Morris must have forgiven Robert's father, William Heysham, for his role in investigating Morris for the Poleacre Victorious scandal.

n addition to his government job, Robert may have had a commercial or private relationship with Robert Morris.
10 May 1784. "Articles of Partnership of Robert Morris, Gouverneur Morris and William

Robert was in the finance office until 1785. What he did from then until he got a federal job in 1789 I do not know. Robert had been earlier identified as a merchant, probably working with his father. While he had other occupations throughout his life, he may have kept his hand in as a merchant. Note that his widow, Mary, had a Drygood store at 48 North 6th street in the 1820's. It is likely this was an inheritance from Robert, vice a new start-up.

In the 1790 census of Philadelphia there were three men living in the household of William Heysham at 107 Arch Street. These were probably William Sr., William Jr., and Robert.

In 1791 Robert Heysham received, for his pay, $258.00 - from "Journal: 1st-13th Congress." He was still the engrossing clerk at this time. Robert Heysham (Haysham) then became the "principal clerk, office of secretary of U.S. Senate" replacing Benjamin Bankson - from "The Papers of Alexander Hamilton." He appears to have held this job from as early as the summer of 1791 to as late as 1803.

In 1793 a Robert Heysham owned 400 acres of land in Northampton township. This may be a 'land deal' of Captain William Heysham's son or a previosuly unknown son of Thomas Heysham of Lower Smithfield township, Northampton county.
Theodore Banta, in "The Sayre Family," says "He [Robert Heysham] acted as adjutant under Gen. William McPherson, during the insurrection in Western Pennsylvania."

Robert became the Deputy Naval Officer [of Customs] for the port of Philadelphia under General William McPherson, his old commander. McPherson was the port's Naval Officer from 1793 to 1813. This was another kind of "spoils system" in which Robert's excellent Federalist connections did him some good.

Robert died on 14 [16] November 1822 in his 63rd year. He was buried at Christ Church. The Will of Robert Heysham of Philadelphia, Pennsylvania, dated 4 July 4 1812, was proved on 16 December 1822.