Research at the moment indicates no direct link between sugars consumption and risk of developing cancer. There is evidence to show that increased body weight, caused by the over-consumption of energy from added sugar foods and drinks, is associated with increased cancer risk.

Key research 

Makarem N, Bandera EV, Nicholson JM, et al. (2018). Consumption of sugars, sugary foods, and sugary beverages in relation to cancer risk: a systematic review of longitudinal studies. Annu Rev Nutr, 38:17-39.
Most studies on total sugar, sucrose and fructose indicated a null association with cancer. An increased association with cancer risk was seen in studies on added sugars and sugary beverages.

Debras C, Chazelas E, Srour B, et al. (2020). Total and added sugars intakes, sugar types, and cancer risk: results from the prospective NutriNet-Sante cohort. Am J Clin Nutr, 112(5):1267-1279.
Total sugar intake was associated with higher overall cancer risk, as well as breast cancer risk. Added sugars, free sugars, sucrose, and sugars fromadded sugars, free sugars, sucrose, and sugars fromsugary drinks, dairy products, and milk-based desserts were associated with increased breast cancer risk. No association was observed for other sugar sources and types.

Other research

World Cancer Research Fund/American Institute for Cancer Research (2018). Continuous update project expert report 2018. Diet, nutrition and physical activity: Energy balance and body fatness.
The Continuous Update Project (CPU) has identified 12 cancers causally linked to greater body fatness. The CUP Panel concluded that, among various other factors, sugar sweetened drinks consumption increases the risk of weight gain, overweight and obesity.

Chazelas E, Srour B, Demetz, et al. (2019). Sugary drink consumption and risk of cancer: results from NutriNet-Santé prospective cohort. BMJ, 366:l2408.
In this large prospective study, the consumption of sugary drinks was positively associated with the risk of overall cancer and breast cancer. 100% fruit juices were also positively associated with the risk of overall cancer. 

Bassett JK, Milne RL, English D, et al. (2020). Consumption of sugar-sweetened and artificially sweetened soft drinks and risk of cancers not related to obesity. In J Cancer, 146(12):3329-3334.
In this Australian cohort there was no association between frequency of consuming sugar-sweetened soft drinks and the risk of cancers, but a positive association was observed for consumption of artificially sweetened soft drinks.

Fiolet T, Srour B, Sellem L. et al. (2018). Consumption of ultra-processed foods and cancer risk: results from NutriNet-Santé prospective cohort. BMJ, 360:k322.
In this large prospective study, a 10% increase in the proportion of ultra-processed foods in the diet was associated with a significant increase of greater than 10% in risks of overall cancer and breast cancer. Diets that include a higher proportion of processed food products tended to be richer in energy, sodium, fat, and sugar and poorer in fibres and various micronutrients.

Wang Z, Uchida K, Ohnaka K, et al. (2014). Sugars, sucrose and colorectal cancer risk: the Fukuoka colorectal cancer study. Scand J Gastroenterol, 49(5): 581–588.
Overall, intakes of sugars and sucrose were not related to colorectal cancer risk either in men or women. The association between sugars intake and colorectal cancer risk differed by smoking status and alcohol use in men, but not in women.