It's the DM's Call
I'm assuming you're using the Spontaneous Divine Casters variant. With that in mind, you're rarely going to find any official rulings on such variants; they are designed to differentiate entire campaigns from traditional campaigns and remain largely unsupported.
That said, the text says that
each time the character gains a new spell level, he gains one or more bonus spells known to add to his list. A cleric may add his two domain spells to his list of spells known, while a druid may add the appropriate summon nature's ally spell to her list of spells known. (An entry of 0 on the table indicates that the cleric knows only his domain spells of that level, and the druid knows only the summon nature's ally spell of that level.)
So I would try to convince the DM that taking an Initiate feat that grants spells is equivalent to gaining the Domain spells, making them available simultaneously with the Domain spells as bonus spells. The character demonstrated his devotion to a deity by taking the feat in the same way he embodies his faith via his chosen Domains. Further, mechanically, the character has expended the resource--an entire feat, one of only maybe seven or eight in his career--so the DM should work with the player so his character enjoys the entire feat's benefits.
A less generous DM would have the spells added to the cleric's spell list as options to be picked normally; such a DM would likely see the value in such additions despite them not being bonus spells because the added spells permit the cleric to employ a wider variety of magic items.
Favored Souls and Initiate Feats
Favored souls (CD 6) use mechanics similar to the spontaneous divine caster variant clerics and druids but favored souls lack the automatic Domain and summon nature's ally spells and have delayed access to higher-level spells. Allowing a favored soul to count his levels as cleric levels for the purposes of Initiate feats is a DM's call in both permission and function, but in the abstract it shouldn't be unbalancing to simply grant the favored soul the Initiate feat's spells as spells known. Personally, I consider the feat Extra Spell (CAr 79) a bad deal ("Just pick better spells!"); getting a handful of new spells known yet not allowing the favored soul to pick the spells seems reasonable.