It's impossible to answer your question without precisely pinning down what you mean by an "embedding." For each monad it is defined for, mfix is required to satisfy three laws: Strictness, Purity, and Left-Shrinking. If you can show that your "embedded" definitions satisfy those, then yes; you do have a valid value-recursion operator. You can find a description of these laws in Chapter 2 of [1]. Since you haven't specified what properties your "embedding" has, it's not possible to say anything in isolation.
More traditionally, adding "new" effects to old monads in Haskell is achieved by using monad transformers [2]. Perhaps that's what you have in mind? If that's the case, then Section 4.9 of [1] lays out how mfix from the base monad can be lifted through a monad transformer stack while preserving the properties. If you can cast your design in terms of MTL style transformers, this would be the way to go. And if you do use the MTL package, then you'll have your MonadFix instances for free, since the library already defines them via the usual instance mechanism.
Cheers,
-Levent.