Constraints on key <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">O</mml:mi><mml:mprescripts /><mml:none /><mml:mn>17</mml:mn></mml:mmultiscripts><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>)</mm

The efficiency of the slow neutron-capture process in massive stars is strongly influenced by neutron-capture reactions on light elements. At low metallicity, $^{16}$O is an important neutron absorber, but the effectiveness of $^{16}$O as a light-element neutron poison is modified by competition between subsequent $^{17}$O$(\alpha,n)^{20}$Ne and $^{17}$O$(\alpha,\gamma)^{21}$Ne reactions. The strengths of key $^{17}$O$(\alpha,\gamma)^{21}$Ne resonances within the Gamow window for core helium burning in massive stars are not well constrained by experiment. This work presents more precise measurements of resonances in the energy range $E_{c.m.} = 612 - 1319$ keV. We extract resonance strengths of $\omega\gamma_{638} = 4.85\pm0.79$ $\mu$eV, $\omega\gamma_{721} = 13.0^{+3.3}_{-2.4}$ $\mu$eV, $\omega\gamma_{814} = 7.72\pm0.55$ meV and $\omega\gamma_{1318} = 136\pm 13$ meV, for resonances at $E_{c.m.} =$ 638, 721, 814 and 1318 keV, respectively. We also report an upper limit for the 612 keV resonance of $\omega\gamma<140$ neV ($95\%$ c.l.), which effectively rules out any significant contribution from this resonance to the reaction rate. From this work, a new $^{17}$O$(\alpha,\gamma)^{21}$Ne thermonuclear reaction rate is calculated and compared to the literature. The effect of present uncertainties in the $^{17}$O$(\alpha,\gamma)^{21}$Ne reaction rate on weak s-process yields are then explored using post-processing calculations based on a rotating $20M_{\odot}$ low-metallicity massive star. The resulting $^{17}$O$(\alpha,\gamma)^{21}$Ne reaction rate is lower with respect to the pre-existing literature and found to enhance weak s-process yields in rotating massive star models.