I don't know the "official" answer to your question, but for words or phrases long enough to achieve respectability I would use near-anagram, partial anagram or quasi-anagram. For example, the rock group Camper Van Beethoven named oone of their recordings "Vampire Can Mating Oven," which is a near-anagram of their name. The smaller the word, the more trivial the near anagram becomes. It would hardly be worth mentioning that "I" is a partial anagram of "tasting" for example.

Cryptic crossword puzzles sometimes use near anagrams or contained words, with clues such as "found in The Senate" (5 letters) [these], or "Two stowed-away in, say, a 747?" [airplane --> airplane --> pair]. I'm not sure if there is any specific name for these contained words, or there is any distinction between words whose letters are already in order and those that are formed by rearranging.