Have found a copy (transcription) of Askham PR on-line. Unfortunately it wasn't very helpful giving no more information about the baptism of Elizabeth Hayton than the IGI entry.
The name Hayton is mentioned several times in the parish register:
John Hayton of Shap married Mary Holme o.t.P. 16 April 1702
William Hayton married Elizabeth Hoggart in 1775. I didn't see a baptism but their daughter Ann was buried in 1793 aged 14 years; her father was described as Wm Hayton of Hilton, Carpenter.
There was also a John Hayton, Weaver at Helton. He married Agnes Hill in 1797 and they had at least two children, Mary in 1797 and John in 1803. Agnes Hayton aged 34 and John Hayton aged 10 days were buried four days apart in April 1803.
John Hayton, widower, then married Ann Mounsey in July 1803.