I agree with the amendment explanation. Mistakes in registers are supposed to be corrected with a numeral in the entry, and the same number in words in the margin, with signature or initials. The sequence of numbers runs all the way through the register book.
In this case, the error seems to be the surname AGNESS, which has been underlined and corrected to HORWOOD. This is confirmed by checking for other entries on the same page - the easiest way is at FreeBMD.
Elizabeth's entry is indexed in Dec qtr 1844, Marylebone, ref. 1/169. FreeBMD shows 8 entries with that reference, one of whom is Sarah AGNESS. It therefore seems likely that the surname has been copied in error from one entry to the next, or possibly the Horwood and Agness informants visited the registrar together and he got their names mixed up.
Either way, you can be confident that Agness is not part of Edward's name.