In short, the answer to your question
can someone become addicted to the absence or the removal of a negative stimulus?
is most definitely yes.
Whilst not intended to be used as “diagnostic criteria” for determining if addiction is present or not, the characteristics of addiction provided through the American Society of Addiction Medicine (ASAM) are widely present in most cases of addiction, and gives an outline of the mechanism required for what you could be talking about, and therefore your hypothesis definitely has legs. You only have to look at alcoholism as an example.
The mechanism works through positive reinforcement, which as you put it, "Well done! here's a candy" in which the "candy", as you will see below, can be pain relief.
The same ASAM link you gave also points out that
Addiction is a primary, chronic disease of brain reward, motivation, memory and related circuitry.
Although experimental studies have produced mixed results, there has been a long-standing belief in the analgesic properties of alcohol and a meta-analysis of controlled experimental studies conducted by Thompson et al. (2016) found that alcohol has the ability to provide a sense of relief from pain.
Alcohol resulted in a small increase in pain threshold.
A moderate-large decrease in pain ratings was also observed.
Higher blood alcohol content is associated with greater analgesia.
Analgesic effects may contribute to alcohol dependence in those with persistent pain.
Alcohol is also a big problem amongst many seeking help from severe stress, depression and/or anxiety through negative urgency which is correlated with greater caudate nucleus activity to alcohol cues (Chester, et al. 2016).
This is because, as pointed out by Sharot (2012)
The caudate nucleus is one of the major targets for the brain’s dopaminergic system (Wise & Rompre, 1989). Dopamine is a principal neuromodulator for reward learning and reward-seeking behaviour (Bayer & Glimcher, 2005; Belin & Everitt, 2008; Schultz, 2001). In particular, dopamine is suggested to signal predictions of rewards, and errors in such prediction–providing a learning signal when predictions do not align with outcomes (Montague, Hymann & Cohen, 2004; Schultz, 2001, 2007).
Through the dependency on the false friend called Alcohol, there is the inability to consistently abstain from seeking relief from pain (emotional or physical) which is part of the impairment in behavioural control. The feelings of escape from physical pain or the emotional attachment to life's hardships, after a while, can lead to craving for the same feeling, either to remove the returning pain, or to prevent its return.
Diminished recognition of significant problems with their behaviours and interpersonal relationships forms another part of the impairment in behavioural control and can arise from this constant "need" for an alcoholic drink, and dysfunctional emotional responses develop.
Chester, D. S., Lynam, D. R., Milich, R., & DeWall, C. N. (2016). Craving versus control: Negative urgency and neural correlates of alcohol cue reactivity. Drug and alcohol dependence, 163, S25-S28. doi: 10.1016/j.drugalcdep.2015.10.036
Sharot, T. (2012). Chapter 3 - Predicting Emotional Reactions: Mechanisms, Bias and Choice. In R. Dolan & T. Sharot (Ed.), Neuroscience of Preference and Choice (pp. 53-72). Academic Press doi: 10.1016/B978-0-12-381431-9.00003-6
Thompson, T., Oram, C., Correll, C. U., Tsermentseli, S., & Stubbs, B. (2017). Analgesic effects of alcohol: a systematic review and meta-analysis of controlled experimental studies in healthy participants. The Journal of Pain, 18(5), 499-510. doi: 10.1016/j.jpain.2016.11.009