[The Saucepan Man]

Tolkien frequently refers, in his Letters, to the Ainur as 'angelic powers' and also as 'gods', although they were of a lesser order than Iluvatar, who was the one God. What power they have is delegated from Iluvatar and to him alone is reserved the power of creation.

This is perhaps best explained in Letter #131 (dating from late 1951):

The cycles begin with a cosmogonical myth: the Music of the Ainur. God and the Valar (or powers: Englished as gods) are revealed. These latter are as we should say angelic powers, whose function is to exercise delegated authority in their spheres (of rule and government, not creation, making or re-making). they are 'divine', that is, were originally 'outside' and existed 'before' the making of the world. Their power and wisdom is derived from their Knowledge of the cosmogonical drama, which they perceive first as a drama (that is as in a fashion we perceive a story composed by some-one else), and later as a 'reality'.

Later in the same letter he continues:

The Knowledge of the Creation Drama was incomplete: incomplete in each individual 'god', and incomplete if all the knowledge of the pantheon were pooled. For, partly to redress the evil of Melkor, partly for the completion of all in an ultimate finesse of detail) the Creator had not revealed all. The making, and nature, of the Children of God, were the two chief secrets. All the gods knew was that they would come, at appointed times. The Children of God are thus primevally related and akin, and primevally different. Since they also are something wholly 'other' to the gods, in the making of which the gods played no part, they are the object of the special desire and love of the gods.

Notice the difference between 'gods' and 'God'.

In Letter #181 (draft: early 1956), he says:

It is, I should say, a 'montheistic but "sub-creational" mythology'. There is no embodiment of the One, of God, who indeed remains remote, outside the World, and only directly accessible to the Valar or Rulers. These take the place of 'gods', but are created spirits, or those of the primary creation who by their own will have entered the world. But the One retains all ultimate authority, and (or so it seems as viewed in serial time) reserves the right to intrude the finger of God into the story: that is to produce realities which could not be deduced even from a complete knowledge of the previous past, but which being real become part of the effective past for all subsequent time (a possible definition of a 'miracle').

As for the peoples of Middle-earth, it seems that, to the extent that they were aware of Iluvatar and the Ainur, they viewed them much in this way. So, Iluvatar was seen as the one God, but the Valar were powers who could be called upon in time of need. This is reflected, for example, in the cries of Legolas and Frodo to Elbereth. Similarly, Faramir's Men offer up a silent prayer to the West at Henneth Annun. Elves and Numenoreans, by virtue of their lore (and, in the case of some Elves, personal experience) had the greatest knowledge in this regard, while other Men, such as the Rohirrim, and Hobbits (Bilbo, Frodo et all being exceptions) appear to have had little conception of the 'Lords of the West' or of Iluvatar.

Hope this helps.