While light-matter interaction mediated by strong exciton-plasmon coupling has been demonstrated to increase the absorbance and spontaneous emission in coupled transition-metal dichalcogenide and metal nanostructures, incorporating it in an optoelectronic device in a controlled manner is challenging. Here we report that the exciton-plasmon coupling can be tuned in a chemically synthesized hybrid of ${\mathrm{WS}}_{2}$ and $\mathrm{Ag}$ nanoparticles (NPs) capped with an insulating polyvinylpyrrolidone (PVP) layer by controlling the coverage area and the PVP layer thickness. Furthermore, we decorate large-area single-layer graphene with these nanostructures to create the hybrid channel for a three-terminal phototransistor. The fabricated device exhibits superior gate tunability and extremely high photoresponsivity (up to 3.2 \ifmmode\times\else\texttimes\fi{} ${10}^{4}$ A/W), which is more than 5 times higher than for the bare $\mathrm{graphene}$/${\mathrm{WS}}_{2}$ hybrid device, along with a low noise equivalent power (approximately ${10}^{\ensuremath{-}13}$ $\text{W}/{\text{Hz}}^{0.5}$) and higher specific detectivity of approximately ${10}^{10}$ Jones units in a wide wavelength range (325--730 nm). The additional PVP capping of $\mathrm{Ag}$ NPs helps to suppress the direct charge and heat transfer and, most importantly, increases the device stability by preventing the degradation of the ${\mathrm{WS}}_{2}$-$\mathrm{Ag}$ hybrid system. Our work demonstrates a strategy towards obtaining an environmentally friendly, scalable, high-performance broadband phototransistor by tuning of the exciton-plasmon coupling for next-generation optoelectronic devices.