〜を押して
despite (and pushing through)
Noun + を押して
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を押して means doing something despite an obstacle, by forcing your way through it: 彼は病気を押して、会議に出席した, "despite being ill, he forced himself to attend the meeting".
The noun is usually your own physical condition (病気, 熱, 痛み, けが) or other people's objections (反対). Occasionally it's the weather: 悪天候を押して. The second half is what you did anyway, often out of duty or determination.
押す means "to push", so it's literally "pushing it aside". It's close to をものともせず, but を押して stresses the effort and strain, while をものともせず stresses that the obstacle didn't matter.
Don't confuse it with を通して, "through (a channel)".
Examples
彼は病気を押して、会議に出席した。
Despite being ill, he forced himself to attend the meeting.
二人は周囲の反対を押して、結婚した。
The two of them married in the face of everyone's objections.
熱を押して、試験を受けた。
I took the exam despite having a fever.
足の痛みを押して、最後まで走り抜いた。
She pushed through the pain in her leg and ran to the end.
悪天候を押して、捜索が続けられた。
The search continued despite the bad weather.