This paper reports on the evaluation of a polyvinyl alcohol (PVA) "twin-chain" polymer network (TC-PN) combined with an oil-in-water nanostructured fluid (NSF) for the removal of a polyvinyl acetate (PVAc) varnish. Small Angle X-ray Scattering, Confocal Laser Scanning Microscopy, and Fluorescence Correlation Spectroscopy showed that the structure of the gel and the NSF are only minimally altered by loading the fluid into the gel. The NSF is partially free to diffuse through the network, but also interacts with the gel walls. During the cleaning, the dynamics of the fluid at the gel-substrate interface are controlled by the osmotic balance taking place among the interconnected pores. These features grant effective and controlled cleaning performances. The case study identified for this research is Pablo Picasso's The Studio (L'Atelier, 1928), one of the masterpieces in the Peggy Guggenheim Collection, Venice (PGC). In 1969 the oil painting, originally unprotected, was wax-lined and then varnished using a PVAc varnish. Over the years, the white shades of the painting have been compromised by the yellowing of the varnish and soiling of deposits. On painting mock-ups, the NSF-loaded hydrogels allowed the swelling and softening of PVAc varnish and wax layers, which were then removed with gentle mechanical action. Effective varnish and wax removal at the micron scale, and the absence of residues from the cleaning system (gel and NSF), were confirmed by Fourier Transform Infrared Spectroscopy (FTIR) 2D imaging. The effective and safe removal of the aged PVAc varnish and wax layer from the surface of the painting was then carried out using the same cleaning protocol successfully tested on the mock-ups, setting the NSF-loaded PVA TC-PNs as robust and reliable tools for the cleaning of sensitive works of art.