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We present JWST NIRSpec prism spectroscopy of gravitationally lensed galaxiesat $z\gtrsim9$ found behind the massive galaxy cluster Abell 2744 in theUNCOVER Cycle 1 Treasury Program. We confirm the source redshift via emissionlines and/or the Ly$\alpha$ break feature for ten galaxies at z=8.50-13.08 downto $M_{\rm UV}=-17.3$. We achieve a high confirmation rate of 100\% for $z>9$candidates reported in Atek et al. (2023). Using six sources with multipleemission line detections, we find that the offset of the redshift estimatesbetween the lines and the Ly$\alpha$ break alone with prism can be as large as$\pm0.2$, raising caution in designing future follow-up spectroscopy for thebreak-only sources. With spec-$z$ confirmed sources in UNCOVER and theliterature, we derive lower limits on the rest-frame ultraviolet (UV)luminosity function (LF) at $z\simeq9$-12 and find these lower limits to beconsistent with recent photometric measurements. We identify at least twounambiguous and several possible active galactic nucleus (AGN) systems based onX-ray emission, broad line (BL) H$\beta$, high ionization line (e.g., NIV]1487,CIV1549) detections, and excess in UVLF. This requires the AGN LFs at $z\simeq$9-10 to be comparable or even higher than the X-ray AGN LF estimated at$z\sim6$ and indicates a plausible cause of the high abundance of $z>9$galaxies claimed in recent photometric studies may be AGNs. One UV-luminoussource is confirmed at the same redshift as a dusty BL AGN at $z=8.50$ with aphysical separation of 380 kpc in the source plane. These two sources showblueward Ly$\alpha$ line or continuum emission, suggesting that they reside inthe same ionized bubble with a radius of $7.69\pm0.18$ pMpc. Our results implythat AGNs have a non-negligible contribution to cosmic reionization.

UNCOVER: A NIRSpec Census of Lensed Galaxies at z=8.50-13.08 Probing a High AGN Fraction and Ionized Bubbles in the Shadow