Q: Using the indefinite article before "rain" I have recently learned the use of the indefinite article before uncountable nouns to talk about an unspecific instance. Can I use "a heavy rain" in the following sentence to communicate that I am talking about an instance of heavy rain (a heavy downpour)?

The water level in the lake is much higher after a heavy rain.


A: Rain may be countable or uncountable:

1988   R. Rendell Veiled One (1989) xiv. 188   Rain was streaming down the window, making the glass opaque.


2006   Africa News (Nexis) 27 Mar.   The current Inter-Tropical Convergence Zone (ITCZ) south of the Equator is likely to bring about more good rains over the country during the next three weeks.

"Rain", in common with most weakly uncountable nouns, if preceded by an adjective - a + adjective + rain - then the adjective has a partitive effect:
A noun, when used uncountably, describes a set of homogeneous items. However, that set has subsets (i.e. types), e.g. "A knowledge of French"; "A warm rain fell"; "A deep sorrow affected him", etc.
The use of "a/an" is licensed in a/an's meaning of "one example of"; "a type of".
The OED puts this as

I. Indicating indefiniteness.

*

*Used in an indefinite noun phrase referring to something not specifically identified (and, frequently, mentioned for the first time) but treated as one of a class: one, some, any (the oneness, or indefiniteness, being implied rather than asserted).

a. Before a noun denoting an individual object or notion, or denoting an individualized substance, quality, or state, and before a collective noun.
1847   H. W. Longfellow Evangeline i. i. 59   A celestial brightness—a more ethereal beauty.


A: Lexico doesn't define rain as "a rainfall" and it gives no examples of that usage, though it does define rains as "falls of rain".
"a rain" is uncommon, especially in the UK, though a heavy rain and a light rain are not.
Google Books shows a rain of (eg manna from Heaven) to be more common than just a rain. Dickens uses only the former:

...desk bespattered with a rain of ink. [Bleak House]

Though Charlotte Brontë's Jane Eyre has:

...the cold winter wind had brought with it clouds so sombre, and a
rain so penetrating, that further out-door exercise was now out of the
question.

A search shows that, even 60 years since he wrote the song, Bob Dylan still owns the phrase A Hard Rain, but your "a heavy rain" is certainly public domain.

A: I do agree with @OLd Brixtonian. I just wanted to add that your sentence may mean different things depending on whether you use the article a or not. And there are many contexts where uses with or without the article are interchangeable.

The water level in the lake is much higher after a heavy rain [particular instance or after every heavy rain].


The water level in the lake is much higher after heavy rain. [every time it rains heavily, does not refer to a particular instance]

Compare:

Swimmers are being warned to stay out of the sea for at least 48 hours
after heavy rain, due to pollution fears. (Newstalk)

and

The Met Office has warned there could be some damage to property and travel disruption in the event of a heavy rain. (express.co.uk)

Usually, heavy rain (without a) is used more commonly in weather forecasts.

A: This question is based on an erroneous assumption - that rain is never countable.

In many tropical countries the produce of the land in grain or roots, during the whole year, depends upon one rain in the spring. (Familiar Letters on Chemistry; Liebig, Justus, Freiherr von; 1843)
In the drought-disaster areas, everyone knew that one rain does not break a drought, but farmers and townsfolk alike drew a deep, fresh breath and hoped. (Rain; TIME, 1953)
But since February, most of the county has seen only two rains ,
each registering less than half an inch. (FOCUS ON DROUGHT IN THE
HEARTLAND; Jim Yardley; Atlanta Journal Constitution, 1996)
Right now, two rains into the rainy season, there's enough water
and food for the group to travel together, but just barely. (Almost
Human; Mary Roach; National Geographic, April 2008)
Since then, there have been three rains, followed instantly by prayers
of thanks. But now the dry season begins. (Echoes of the Great Dust
Bowl; Hugh, Sidey; TIME, 1996)

Though the non-count use may be the more common by far, the count use of rain seems well established, even without a modifier. In the right context, it's perfectly natural.
The meaning of rain in the above is simply a single instance of rainfall. From the American Heritage Dictionary,

rain
n.
1.
a. Water condensed from atmospheric vapor and falling in drops.
b. A fall of such water; a rainstorm.

