What's the meaning of "Make a play for it"?

Question

I checked the phrase in Cambridge Dictionary and it means

"to try to get something, or start a relationship with someone, sometimes by using a plan" but I don't now what the meaning is in this case.So,

Could you please explain what the meaning of "Make a play for it" is in this text?

The full text is:

That evening, Shawn saddled a new horse, a copper-coated mare, for the first time. She was ready for a short ride, Shawn said, so we mounted, him on the mare, me on Bud. We made it about half a mile up the mountain, moving deliberately so as not to frighten the horses, winding our way through the wheat fields. Then I did something foolish. I got too close to the mare. She didn’t like having the gelding behind her, and with no warning she leapt forward, thrusting her weight onto her front legs, and with her hind legs kicked Bud full in the chest. Bud went berserk. I’d been tying a knot in my reins to make them more secure and didn’t have a firm hold. Bud gave a tremendous jolt, then began to buck, throwing his body in tight circles. The reins flew over his head. I gripped the saddle horn and squeezed my thighs together, curving my legs around his bulging belly. Before I could get my bearings, Bud took off at a dead run straight up a ravine, bucking now and then but running, always running. My foot slipped through a stirrup up to my calf...My instincts told me to let go of the saddle horn—the only thing keeping me on the horse. If I let go I’d fall, but I’d have a precious moment to reach for the flapping reins or try to yank my calf from the stirrup. Make a play for it, my instincts screamed. Those instincts were my guardians. They had saved me before, guiding my movements on a dozen bucking horses, telling me when to cling to the saddle and when to pitch myself clear of pounding hooves. They were the same instincts that, years before, had prompted me to hoist myself from the scrap bin when Dad was dumping it, because they had understood, even if I had not, that it was better to fall from that great height rather than hope Dad would intervene. All my life those instincts had been instructing me in this single doctrine—that the odds are better if you rely only on yourself. Bud reared, thrusting his head so high I thought he might tumble backward. He landed hard and bucked. I tightened my grip on the horn, making a decision, based on another kind of instinct, not to surrender my hold. Educated by Tara Westover

Answer 1

To make a play (for something) is an idiomatic expression that means to try and achieve some difficult objective. Example:

The batter swings! It's a looping grounder to far right field! He's rounded first and is making a play for second ... Safe!

In your example the narrator must first let go of the saddle horn and risk falling from the horse, in order try to reach the reins and retake control.

Answer 2

"to try to get something" - to attain a goal (of reaching the reins or freeing her boot from the stirrup, and thus remaining on the horse, despite its bucking).