"This isn't fucking fair." Those are the four words from Katherine Soper's Bruntwood Prize-winning play that stick. Zero hours contracts aren't fair. Disability benefits cuts aren't fair. Soper's teenaged protagonist, Tamsin (Erin Doherty), has to contend with both. A carer for her younger brother Dean (Joseph Quinn), whose employment and support allowance has just been cut, we meet her as she begins working shifts in a soulless packing warehouse. The slogan at the door reads, mockingly, "work, enjoy, improve".

The play's title, suggestive of Amazon's wishlist (though Soper is careful not to make this about any one company), also nods to the unfulfilled wishes of its characters. There is, at times, a melancholy sense of what might have been, or what could be under different circumstances. Never is this more acute than in Tamsin's conversations with upbeat, puppyish fellow warehouse worker Luke (Shaquille Ali-Yebuah), whose entrance into her life suggests new - if distant - possibilities.

Like Alexander Zeldin's hyper-naturalistic play Beyond Caring, which offered a window onto the shifts of a group of zero hours cleaners, Wish List makes visible the hidden, atomised labour that oils the wheels of consumer capitalism. We all know that the supply chains behind our Amazon deliveries are far from pretty, but it's another thing actually seeing it. As one of Soper's characters suggests, it's all a question of perspective.

The perspective offered by Matthew Xia's production is that of the warehouse floor. Ana Inés Jabares-Pita's design collapses home and workplace, cramming Tamsin and Dean's small kitchen and bathroom into a corner of the otherwise industrial space. It's a neat visual metaphor for the insidious encroachment of work into every area of people's lives. Tamsin's barely had her morning cup of tea before she's on the packing line, frantically assembling boxes. Her home life, meanwhile, lingers at the edge of the working day, always present.

In this hostile world of precarious labour, orders come – like the many boxes of Jabares-Pita's set – from above, heedless of their impacts. Soper draws an astute but depressing comparison between the inflexibility of the warehouse's rules and the inflexibility of fit-for-work assessments. Neither system is built for individual, fallible human beings. All the while, Giles Thomas's sound design queasily evokes the race to meet targets and the throbbing tension of a life in which something, somewhere, has to give.

There's no doubt that these are lives that need seeing, injustices that need protesting. This isn't fucking fair. Wish List, though, lacks the unadorned precision of a show like Beyond Caring. There are moments in the play when you can almost hear the cogs whirring, forcing plot turns of mini revelation or surprise reversal. At one point in particular, the implicit is made unnecessarily explicit, as a marginal character suddenly turns into a vessel for Soper's message.

But what can't be shaken – by either narrative missteps or corporate greed – is the hopeful tenderness of these characters. As delicately and compassionately portrayed by Doherty, careworn Tamsin in particular is all heart. In one of the show's standout scenes, Luke persuades her into a tentatively joyful rendition of Meatloaf's "I'd Do Anything For Love" – a power ballad that's all cheese, but with generous dollops of feeling. It's that unapologetic emotion that feels like the core of the play and, perhaps, the antidote to the unfeeling systems it exposes.

Wish List runs at the Royal Exchange, Manchester until 15 October.