|I think that you have got something wrong with your dates. He died in 1821 and fathered an illegitimate child in 1831, it may be possible now but not then.|
You also say that the illegitimate child was named Richard Sharpe. If a child was illegitimate they are usually given the mother's name at birth or perhaps the couple were either married or considered themselves married under common law at the time of birth. Possibly the mother had the maiden name Sharpe or was married to another person named Sharpe or perhaps the child Richard adopted his father's name later in life, although unusual as he would have had little if any knowledge of his father. I presume that you must have some knowledge of his mother, her name, place of birth and date of birth would be of assistance in locating the correct Richard Sharpe. Also where was the child born.