This variant concerns the name of the man who was Abijah's son and Jehosaphat's father: the Alexandrian Text (allied with family-1, 700, the early Sahidic version, and some Old Latin copies) supports "Asaph," but the Byzantine Text (allied with L, W, the Vulgate, the Sinaitic Syriac, the Curetonian Syriac, and the Peshitta) supports "Asa." According to the Old Testament, the king's name was Asa; Asaph was a psalm-writer contemporary with David. It seems unlikely that Matthew would confuse these two individuals.
The reading "Asaph" may be an early copyist's quirky attempt to enhance Jesus' genealogy by removing the name of Asa with the name of the more spiritually accomplished Asaph. (The same quirk may be at work in verses 10-11, where, in the Alexandrian Text, the name of king Amon is replaced by the similar name of the prophet Amos.)
A parableptic error may also explain the Alexandrian reading: if an inattentive early copyist's line of sight momentarily wandered to the occurrence of "-sapha-" in the name "Jehosaphat" in the following line, he may have added the "ph," repeated the name the same way without consulting his exemplar, and then resumed writing at the correct place, never noticing his mistake.
Subsequent copyists, rather than viewing "Asaph" as a mistake, may have considered it a spelling-variation. Several of the names in the genealogy are the subjects of orthographic variation: in verse 2, Aleph has "Isak" instead of "Isaak, and in verse 3, B has "Zare" instead of "Zara." Aleph and B both have "Boez" and "Iobed" instead of "Booz" and "Obed" in verse 5. Aleph spells Solomon's name as "Salomon" in verse 6. In verse 9, Aleph has "Achas" twice, instead of "Achaz." In verse 10, B names "Amnon" as the son of Manasseh, but then immediately calls him Amos. In verse 12, B twice spells Salathiel as "Selathiel." In verse 13, Aleph names "Abiout" instead of "Abioud," and in verses 14 and 15, Aleph names "Eliout" instead of "Elioud." All this goes to show that in the Alexandrian channel of transmission, there was wide latitude in the spelling of proper names -- in which case, the name "Asaph" may have originated as a localized spelling-variation, rather than as an erroneous reference to the psalm-writer.
This is a straightforward contest between "David" and "David the king," where David is mentioned the second time in the verse. The Byzantine reading "David the king" constitutes a case of scribal conformation; a natural desire to consistently repeat the names in the genealogy led an early copyist to repeat David's title as well. The Byzantine reading here is supported by the Vulgate (although some copies of the Vulgate diverge). Support for the reading "David" (without "the king") is represented by Papyrus-1 (from the 200s), and by Aleph, B, 1, 1582, 700, the medieval uncial S, and by the Sinaitic Syriac, the Curetonian Syriac, the Peshitta, and the early Sahidic version.