Burke's is generally reliable; are you sure it isn't just a misleading pronoun? Having 3 Francis Winningtons in succession is bound to be a bit confusing. A web page I've found showing the correct arrangement claims to be sourced to Burke's of 1934. The following is the tree as best I can make out:

Sir Francis Winnington
Salwey Winnington
Thomas Winnington
Francis Winnington, of Broadway
Francis Winnington (MP)
Edward Winnington
Sir Edward Winnington, 1st Baronet and descendants
Edward Jeffreys
Yours,