Tim Powys and others think this John m. Anne Willoughby.[1]
A book about the Berkeley family also says he was heir apparent to Richard and that the couple got the manor of Erlingham for their lives.[2] If they did, no surviving issue, as Richard's sisters were co-heiresses.
According to Wikipedia, Richard Beauchamp married Elizabeth, daughter of Humphrey Stafford of Grafton and they had Sir John, died young; Elizabeth (d.1503), married, Robert Willoughby, 2nd Baron Willoughby de Broke (d. 1521), Anne, married, Thomas Lygon, of Madresfield, Worcs and Margaret, married, William Rede of Gloucester (father of Richard Rede/Reade).[3]
According to Wikipedia, unsourced, John is the possible son of Richard Beauchamp K.B. (abt.1435-1503) and Elizabeth (Stafford) Beauchamp (abt.1425-) and died young. They were detached as his parents until a better source can be found linking him to these parents.
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Richardson does mention a John Beauchamp Knt who married Anne Willoughby but does not explain who he was.