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Time-resolved photometry is an important new probe of the physics of condensate clouds in extrasolar planets and brown dwarfs. Extreme adaptive optics systems can directly image planets, but precise brightness measurements are challenging. We present VLT/SPHERE high-contrast, time-resolved broad H-band near-infrared photometry for four exoplanets in the HR 8799 system, sampling changes from night to night over five nights with relatively short integrations. The photospheres of these four planets are often modeled by patchy clouds and may show large-amplitude rotational brightness modulations. Our observations provide high-quality images of the system. We present a detailed performance analysis of different data analysis approaches to accurately measure the relative brightnesses of the four exoplanets. We explore the information in satellite spots and demonstrate their use as a proxy for image quality. While the brightness variations of the satellite spots are strongly correlated, we also identify a second-order anti-correlation pattern between the different spots. Our study finds that KLIP reduction based on principal components analysis with satellite-spot-modulated artificial-planet-injection-based photometry leads to a significant (∼3×) gain in photometric accuracy over standard aperture-based photometry and reaches 0.1 mag per point accuracy for our data set, the signal-to-noise ratio of which is limited by small field rotation. Relative planet-to-planet photometry can be compared between nights, enabling observations spanning multiple nights to probe variability. Recent high-quality relative H-band photometry of the b–c planet pair agrees to about 1%. [ABSTRACT FROM AUTHOR]