The phrase a sprat to catch a mackerel, and its variants, denote a small outlay or risk ventured in the hope or expectation of a significant return.

This is a metaphor from fishing, in which sprats are used as bait to catch larger fish.

Interestingly, in A Collection of English Proverbs Digested into a convenient Method for the speedy finding any one upon occasion; with Short Annotations. Whereunto are added Local Proverbs with their Explications, Old Proverbial Rhythmes, Less known or Exotick Proverbial Sentences, and Scottish Proverbs (Cambridge: Printed by John Hayes, Printer to the University, for W. Morden, 1670), the English naturalist and theologian John Ray (1627-1705) quoted a proverbial phrase expressing the reverse notion:

To fish for a herring and catch a sprat.

These are, in chronological order, the earliest occurrences of the phrase a sprat to catch a mackerel, and variants, that I have found:

1-: From Fortune’s Tricks in Forty-Six. An Allegorical Satire (Dublin: Re-Printed by R. James, 1747):

For. If you can keep your Temper a Moment, I will let you into a Secret, that shall appease this Gust of Resentment in you.
Capt. Madam, I am calm.
For. Know then, that the Ministry, in what they have done, have only thrown out a Sprat to catch a Salmon. Under Colour of this seeming Parsimony, the Supplies went smoothly, and liberally on; for who cou’d refuse any thing to so gracious a Monarch, who parts with his favourite Household for the Ease of his People?

2-: From The Eccentricities of John Edwin, Comedian. Collected from His Manuscripts, and Enriched with Several Hundred Original Anecdotes. Arranged and Digested by Anthony Pasquin, Esq. (London: Printed for J. Strahan, [1798])—John Edwin (1749-1790) was an English actor; Anthony Pasquin was the pseudonym of the Irish painter and engraver John Williams (died 1818):

“Give a sprat to catch a salmon.”

About twelve years ago, Mr. Vikery, and his fragrant bear’s grease, made no inconsiderable noise in the metropolis—when I was editor of a certain morning paper, he came to the office on a Friday evening, to have his advertisement inserted in the print of the succeeding day—It is customary, when the compositors are deranged by such requests being made at such an unusual hour, to reward them with some douceur—Vikry [sic], being distrustful, remained in the office until he saw his desires fulfilled, and then shabbily quitted the men, without even thanking them for their extraordinary efforts to oblige him—However, in this instance at least, the dealer in fat was unwise, for he had scarce left the place before they inserted an L instead of an R, in the second word, and it ran as follows:
                    Vikery’s Flagrant Bear’s Grease.
As this seemed like the casual admittance of light into the dark caves of imposition, the world laughed—and the butcher of bears hung his head in dismay!

3-: From a comment, by the English author and political reformer William Cobbett (1763-1835), about a newspaper article quoted in his book, Porcupine’s Works: Containing Various Writings and Selections, Exhibiting a Faithful Picture of the United States of America (London: Printed for Cobbett and Morgan, 1801):

This I take to be a lure to inveigle the Americans from home, and to induce them to sail from the West Indies without convoy. The plundering villains had been ordered to let a vessel or two of little value escape, that the report might spread through the continent, and bring them better prey. It was, as they call it, donner un œuf pour avoir un bœuf *, or, in the language of Christians, throwing a sprat to catch a herring.

* French donner un œuf pour avoir un bœuf translates as to give an egg to get an ox. (Incidentally, the French proverb qui vole un œuf vole un bœuf, literally who steals an egg steals an ox, means once a thief, always a thief.)

4-: From “There is a Secret, Find It Out!” A Novel (London: Printed at the Minerva-Press, for Lane, Newman, and Co., 1808), by the English author Elizabeth Meeke (1761-1826?):

John […] even went so far as to say, “that if the squalling fellow would meet him half way, he had no objection to being friends with him, since it must make it disagreeable to the family their being at variance, and as he had been kind to his father, though he made no doubt he had been baiting with a sprat to catch a herring, but that was of little consequence.”

5-: From The Tarantula; Or, The Dance of Fools. A Satirical Work (London: Printed for J. F. Hughes, 1809), by the Irish author Eaton Stannard Barrett (1786-1820):

Another plan is started which promises a plentiful harvest to the projectors: two shops are opened, in the same street, which outvie any thing I have yet stated in bargains.—They purport to have no connection with any other shop—to sell silk at the price of dowlas—“filthy dowlas”—and, for ready money only, to let you have five guineas worth of goods for one golden picture of his present majesty—God bless it!—and, in some instances, they keep their word,—but it is only throwing away a sprat to catch a mackrel [sic].

6-: From Hamlet Travestie: In Three Acts. With Annotations by Dr. Johnson and Geo. Steevens, Esq. and Other Commentators (London: Printed for J. M. Richardson, 1810), by the English playwright John Poole (1786-1872):

Enter Horatio.
Hamlet. Horatio, is that you? I’m glad to meet you.
Horatio. My honour’d lord, most proud am I to greet you.
Hamlet. Horatio, you’re as tight a lad, I say,
As one may meet with in a summer’s day.
Horatio. Come, that won’t do, my lord:—now that’s all gammon.
He’s throwing out a sprat to catch a salmon.

The English author Charles Reade (1814-1884) used the phrase with whale in “It Is Never Too Late to Mend.” A Matter of Fact Romance (London: Richard Bentley, 1856):

‘Come,’ said Robinson, ‘here is a spot that looks likely to a novice: dig and cut it up all you can.’
George was mystified but obeyed, and soon the place looked as if men had been at work on it some time. Then Robinson took out a handful of gold-dust and coolly scattered it over a large heap of mould.
‘What are you at? Are you mad Tom? Why there goes five pounds. What a sin!’
‘Did you never hear of the man that flung away a sprat to catch a whale?’