Does the 2nd Lvl slot given by Precocious Apprentice qualify the character for bonus spells of second level with a ability score of 14 or higher in the corresponding ability?
2nd-level bonus spells aren't gained solely from having a high ability score and taking Precocious Apprentice
Abilities and Spellcasters, in part, says
The ability that governs bonus spells depends on what type of spellcaster your character is: Intelligence for wizards; Wisdom for clerics, druids, paladins, and rangers; or Charisma for sorcerers and bards. In addition to having a high ability score, a spellcaster must be of high enough class level to be able to cast spells of a given spell level.… For instance, the wizard Mialee has an Intelligence score of 15, so she’s smart enough to get one bonus 1st-level spell and one bonus 2nd-level spell. (She will not actually get the 2nd-level spell until she is 3rd level wizard, since that’s the minimum level a wizard must be to cast 2nd-level spells.) (Player's Handbook 7—8 and emphasis mine)
Despite the feat Precocious Apprentice granting the creature "an extra 2nd-level spell slot that must be used initially to cast only the chosen spell" (Complete Arcane 181), as illustrated above until the creature's class levels in its arcane casting classes allow it to cast 2nd-level spells, the creature just doesn't get 2nd-level bonus spells due to a high ability score.