The 1900 Census of East Union Township, Worth County, Missouri
Thomas McCrery, born June 1861 in Illinois, widower, farmer
http://www.predinkle.com/census/1900-census/1900indexall.php
This census was enumerated in June 1900, before Alfred was born in September 1900. So, while this "appears" to be the correct Thomas McCrery, in the potentially correct County (Andrew Jackson McCrery died in this county in 1903; Thomas was found here in the 1910 Census), born in the correct state of Illinois; but, is this another person? If it IS the correct one, was this one just "claiming" to be a widower and had run out from his wife in another county (Nodaway?) ? And/or is the birth date of Alfred incorrect and was born in 1899? The 1910 census shows Alfred to be 10, born in Missouri to both parents born in Missouri. This would "indicate" Alfred may have been born to another family and was adopted/otherwise brought into the family. OR, his father was a widower in June 1900 and fathered Alfred by September, but had not married her or the mother shown on the 1910 census WAS his mother. Questions, Questions...

The Worth County Birth Index runs 1883-1894, to early. The Death Index runs 1883-1894, then 1910 to 1957.

This site has a LOT of great genealogy information...but does not address the answer to the questions:
http://www.predinkle.com/