Heavy-handed is the opposite of subtle. Even by its dictionary definition, it can take two forms:
- It can mean being clumsy - imagine trying to do delicate work when your hands are very heavy.
- It can mean using blunt force - imagine bringing your hands down heavily on somebody else, e.g. "He ruled the kingdom with a heavy hand."
The two shades of meaning are not identical, but their result can be quite similar: in the case of writing, it means that the writing is blunt and clumsy. It feels like its elements were forced into place - whether because the author didn't know how to do better, or because he was determined to reach some particular goal and didn't care whether that fit organically with the story.
"Heavy-handedness" can apply to many different elements and at many different resolutions. So if you have three long paragraphs introducing your antagonist and saying how stupid his ideology of gluten-hating is, that might be a heavy-handed description. But if the entire plot takes place in a gluten-free dystopia, and everybody who's opposed to gluten is presented as cruel and malicious, and it just so happens that the heroes wind up saving the day and defeating the dragon by baking the very first loaf of bread in a hundred years - then it's your plot and your setting that are heavy-handed.
Obviously, in the first case, you just edit the three paragraphs.
In the second case, if you want to avoid heavy-handedness, that means you want to introduce subtlety and nuance. Instead of the bad guys being cookie-cutter evil bastards, you'll want to introduce shades of grey, variety of opinion, justifications (flawed, but compelling) for their beliefs and actions.
Another form of heavy-handedness in plot is when it feels very obvious that the writer is forcing some kind of plot or outcome. Gilligan can never get off the island; Voyager can never find a shortcut back home. Or when the motivation just seems blunt and vulgar - for example, killing or kidnapping the protagonist's wife or children can often feel heavy-handed, because it's latching on to a generic, cliche target we know is supposed to be important (even if the wife/children are hardly established characters), and using that as a way to control the protagonist and force him to get involved in the actual plot. Or, a "prophecy" that gives the author carte blanche to force the protagonist to do any damn fool thing somebody (i.e.: the author) has written a prophecy about.
Now, a big problem here is that "heavy-handedness" can be very, very subjective. What for one reader might be obvious from page 10 might not be clear to another reader until page 60; in that case, a clear shout-out on page 25 might be "heavy-handed" for the first reader but necessary for the second. Likewise, your pro-gluten friends might love your gluten-free dystopia, feeling this is the most accurate, insightful depiction of society in years, what with the anti-glutenites suppressing free speech and all. And the more genre-savvy you are, the more sensitive you are with cliche plots and "cheap" ways to get the story going, while another reader may enjoy these stories with no difficulty.
So here's how to avoid heavy-handedness in your own writing:
- Have a good pool of readers to rely on for critique and feedback. If they're not bothered by heavy-handedness, you're probably fine.
- Be careful when writing stories with a moral or a specific ideological stance. Such stories can be important and powerful, but with them, you have a specific goal which is distinct and different from organic, natural development of the story. If the goal overrides the story, that's when you're getting heavy-handed.
- Avoid monolithic elements. "Everybody from [PLACE] has [PERSONALITY TRAIT]." "All [GROUP OF PEOPLE] are evil." "[PERSON X] is evil and has absolutely no good qualities whatsoever." "[IDEOLOGY X] is always correct and any divergence from it will end in tears." Instead, look to add nuance, justification, and variety.
- Be genre-savvy. Read a lot in your genre; know the tropes and the cliches. Most particularly, know what genre-savvy people are tired of (e.g., we are tired of women in refrigerators).
- Be aware of what you expect your reader to know, to suspect, to guess at different points in your writing. When you drop clues and foreshadowing, stop and take a good estimate of what kind of reader is likely to figure out what where. When you have a point of information that the reader must understand, avoid simply stating it baldly in dialogue - "Oh my goodness, you're actually my mother!" Instead, look for a way to make the point absolutely clear, while also doing something else which will engage the savvier reader. ("Mom, I've figured out it's you, and I think it's time we started taking your dementia seriously.")
As others have mentioned, heavy-handedness should mostly be a secondary concern to actually getting your story written. It's absolutely fine in a first draft, because clumsy, obvious writing is a really good way to get words on the page and to get your story to the point where it's solid. And since heavy-handedness is subjective (and as a writer, you're likely to be more sensitive to it than plenty of readers), anticipating every possible reaction to every paragraph and every plot element is a lousy strategy. Much better to produce an initial story, and then modify it according to criticism and your own editing skill. (And, if you've developed your own sensitivity to heavy-handedness, your initial drafts might be fine even in their own right.)
Hope this helps, and all the best!