Ann Leckie's The Raven Tower is a short fantasy novel told from the perspective of a demigod. Like all her other novels, it is a deft and thoughtful exploration of a way of being fundamentally different from our own. More relevant to this blog, it is also a compelling account of how deities (and their clerics) in your standard D&D setting might work.
In this article I'm going to try and summarize what gods in The Raven Tower are, and how their powers work. While ultimately I'd suggest you just go read the novel, I understand that not all readers have the time, money, or interest to do so. Still, I recommend it, especially if the breakdown in this post is interesting to you. The post will contain some minor spoilers, although I am confident it will not give away any critical plot details.
Also note that I listened to an audiobook of The Raven Tower, to which I no longer have access, about two years ago. Therefore, my recollections on some points are hazy, and I am unable to go back to the text to clarify.
The nature of gods
Gods in The Raven Tower are beings of pure mind and will. They occupy a particular point in space and can inhabit some physical body (examples include a rock, a dog, a swarm of mosquitos, a skull) but do not need to. Their exact origin, and relationship to the wider cosmology, is left ambiguous in the book. Gods are not omniscient and their awareness, while not strictly limited by senses as we know them, appears limited to their vicinity.
Everything a god says is true. This does not mean that gods cannot lie, but rather that if it speaks an untruth, the god wills that untruth into being. If a god says "that tree is on fire", the tree must either already be on fire, or it catches fire as soon as the sentence is uttered. Gods' speech is not verbal, so much as a pure expression of intent. In this regard, willpower and physical energy appear to be interchangeable, and speech alters the world largely through the application and manipulation of physical laws. Converting one substance into another means rearranging it at an atomic level; causing it to float means applying sufficient force to counter the pull of gravity. I do not remember whether the law of conservation of matter applies in The Raven Tower, but either way, it is certainly easier to manipulate existing matter than to create it fresh.
Speaking into being in this way expends a god's limited supply of consciousness, and so they have to be careful about what they say. Using up too much of its power robs the god of its essence and ability to think or act. I forget whether gods can truly die, but they can come very close by speaking something that exceeds their capacity. For instance, a god can speak a storm into being for a moment, allowing the storm to dissipate naturally afterward; but if a god says a storm will rage on forever, its power will permanently be funneled into keeping that storm going. Thus, the best way to "harm" a god is to trick it into saying something can't afford to make true.
The sociology of religion
Much as gods can expend energy through will, they accumulate energy as it is given to them. Acts that direct energy to a god nourish them; things like prayer are like a little snack, whereas a mass sacrifice, which gives matter and life over to them, is like a hearty meal. The exact mechanics of this exchange are not explored in depth, but the relationship between the direction of energy and the nourishment of gods make for some of the most interesting moments in the novel.
Something like religion, then, tends to emerge in historical relationships between communities of humans (there are no demi-humans in The Raven Tower) and gods. Humans can provide a steady diet of worship and sacrifice, in exchange for various boons, blessings, miracles, etc. At the same time, gods are cautious about what promises they make their worshippers, and often attach terms that allow them to balance their own resources and avoid overcommiting. For example, a god may promise that so long as certain observances are kept up according to its specifications, it will continue to protect a community from eg. drought, or some other narrowly defined threat.
Because gods' awareness is limited, they sometimes make their power "portable" by attaching a promise to a certain person, object, and/or action. This canonically leads to conditions where a blessed object -- a "god-spoken thing" -- will enact some miracle when a command word or gesture is performed, but only once a day.
(I'm extrapolating now, but you have here the basis for your traditional D&D cleric: A person who has adequately demonstrated their fealty is allowed to ask for a certain number of favours per day. Those favours are pre-defined, ranked according to how costly they are to grant -- levels -- and meted out in limited quantities per day. As they continue to prove their worth, clerics are given access to more and better favours.)
Gods have individual personalities and skills. Gods gradually develop techniques to more efficiently enact certain things. The difference could be between, say, a god that provides better crop yields by learning about and subtly accelerating plants' natural growth processes, and one that isn't as knowledgeable about plants and obtains the same effect by providing good weather all year (the latter presumably being a lot more costly). While their relationships with humans can be quite transactional, gods can also grow attached to humans -- or at least their communities, as individuals don't live very long.
The Raven Tower presents a world where gods live in evolving relationships with humans and one another, and where those relationships are embedded in a kind of pseudo-material economy of willpower. The nature and limitations of Leckie's take on fantasy deities corresponds closely to the way they're instantiated mechanically in D&D, making that novel's theology an excellent fit for your kitchen-sink D&D setting.