Research has shown that sugar consumption does not cause hyperactivity or behavioural problems in children. Limited studies suggest a link between sugar-sweetened beverage consumption and risk of ADHD.

Key research 

Wolraich ML, Wilson DB & White JW. (1995). The effect of sugar on behaviour or cognition in children: a meta-analysis. JAMA, 274(20):1617–21.
The meta-analytic synthesis of the studies to date found that sugar does not affect the behaviour or cognitive performance of children. The strong belief of parents may be due to expectancy and common association. However, a small effect of sugar or effects on subsets of children cannot be ruled out.

Other research

Del-Ponte B, Anselmi L, Assuncao MCF, et al. (2019). Sugar consumption and attention deficit/hyperactivity disorder (ADHD): a birth cohort study. J Affect Disord, 243:290-296.
There was no association between always high sucrose consumption between 6 and 11 years and incidence of ADHD, compared with individuals who always presented low consumption.

Flora S, and Polenick C. (2013). Effects of sugar consumption on human behavior and performance. The Psychological Record, 63(3):513-524.
This discussion paper concludes that sugar consumption, even in high amounts, does not contribute to hyperactivity, inattention, juvenile delinquency, reductions in cognitive performance, or other behaviour problems in children or adults. Instead, a high level of glucose, or sugar, consumption actually improves athletic, academic, and cognitive performance, and may enhance self-control.

Kim Y & Chang H. (2011). Correlation between attention deficit disorder and sugar consumption, quality of diet, and dietary behavior in school children. Nutr Res Pract, 5(3):236-245.
There was no significant association between total volume of simple sugar intake from snacks and ADHD development. Children who consumed less sugar from fruit snacks, or whose vitamin C intake was less than the dietary reference intake, were at increased risks for ADHD.

Benton D. (2008). Sucrose and behavioral problems. Crit Rev Food Sci Nutr, 48(5):385-401.
Well-designed studies that have examined the impact of sucrose on the behaviour of children produced no evidence that it has an adverse influence.

Del-Ponte B, Quinte GC, Cruz S, et al. (2019). Dietary patterns and attention deficit/hyperactivity disorder (ADHD): A systematic review and meta-analysis. J Affect Disord, 252:160-173.
Although the current evidence is weak, this study suggests that a diet high in refined sugar and saturated fat can increase the risk of ADHD, whereas a healthy diet, characterized by high consumption of fruits and vegetables, would protect against ADHD or hyperactivity.

Yu C-J, Du J-C, Chiou, H-C, Feng, C-C, et al. (2016). Sugar-sweetened beverage consumption is adversely associated with childhood attention deficit/hyperactivity disorder. Int J Environ Res Public Health, 13(7):678.
There was a dose-response relationship between SSB consumption and ADHD. Children who consumed SSBs at moderate levels and high levels had greater odds of having ADHD, compared with those who did not consume SSBs.

Schwartz DL, Gilstad-Hayden K, Carroll-Scott A, et al. (2015). Energy Drinks and Youth Self-Reported Hyperactivity/Inattention Symptoms. Acad Pediatr, 15(3):297-304.
Risk of hyperactivity/inattention increased by 14% for each additional sweetened beverage consumed. Students reporting consumption of energy drinks were 66% more likely to be at risk for hyperactivity/inattention.