"reckless pursuit" is correct and usable in written English.
You could use it to describe a situation where someone is behaving recklessly in their pursuit of something, such as a goal. For example, "His reckless pursuit of a promotion led him to act in dangerous and unethical ways."
The movie also touches on the reckless pursuit of fame.
This self-described "pollution tourist" seeks a variety of sites facing ruin through human indifference and reckless pursuit of profit.
He injured himself in glorious sporting endeavour; my tumble followed the reckless pursuit of walking on cobbles in high heels.
The central bank should not bankroll government spending, he has said, and a reckless pursuit of inflation will wreck the economy.
The crisis occurred because banks that had grown too big to fail came too close to failure — driven by a reckless pursuit of risk and profit.
If life imitated art, as Wilde insisted in his essay "The Decay of Lying" (1889), he was himself approximating the pattern in his reckless pursuit of pleasure.
I silently thanked God that my instructors at the Skip Barber Racing School were watching over my reckless executive pursuit from the edges of the track.
In retrospect, that vision of absolute nothingness was a fitting image for the reckless executive pursuit into which I had just lost myself.
Jack Aubrey's ship is part of a larger bureaucratic system that defines the limits of his autonomy and the particulars of his duty; but he is, within these parameters, free to be creative, even somewhat reckless, in pursuit of his designated goals.
More broadly, Germany has been the most fervent voice that reckless economic pursuits should be punished, no matter the human toll.
The incident began around 9 30 p.m. as a suspected drunk- or reckless-driver pursuit in Cudahy by Los Angeles County sheriff's deputies.
