I believe that this was something in one of Conn Iggulden's Conqueror novels, if memory serves correctly. A fun series and a good writer, but he takes a number of artistic decisions for the sake of story telling.
Generally speaking this sounds unlikely, being as Mongol families often accompanied the army on campaign, so there would not be a 'returning,' in quite that context. And when the sources are discussing the spoiled or lazy nature of Mongol rulers, one of the examples they often give is that they spent more time hunting than ruling. For instance, to depict Jochi (eldest son of Chinggis Khan) as lazy and to discredit his successors, after the fall of Gurganj in 1221 he said to have spent the rest of his life hunting in pro-Toluid sources (see Atwood, "Jochi and the Early Campaigns," for more on this matter.) And Kublai Khan was famed for his hunting trips wherein he would sit on the back of four elephants lashed together (his gout being so bad, and him so obese that horse riding was likely very difficult for him as he aged) and shoot at animals released infront of him, as recorded in Marco Polo.
After Temujin, the young Chinggis Khan, lost his father to Tatars and was abandoned by his tribe, he and his family were forced to survive exclusively hunting and gathering what they could, eating such things as mice, dogs, birds or even fishing. That sort of hunting was looked down upon by the Mongols, but generally had positive connotations and was an important means to sustain themselves alongside their herds.
Further, the Mongols and their successors never took to living, except part time, in palaces, commonly setting up gers (yurts in Turkish) in palace gardens or outside of the city. Ogedai Khaan, Chinggis Khan's successor, despite being famed for the construction of Karakorum in Mongolia spent little time living in the city itself, and such customs for the conquerors and their families continued well into the time of Timur and his descendant Babur. Which is not to say they never lived in palaces, but during the height of the united Mongol Empire (1206-1260) palace building and construction never common in Mongolia (though did take place, as not only Karakorum but a number of other small cities and palace like complexes were constructed by Chinggisids, particularly in the Orkhon area).
However, there was, particularly as the 13th century wore on, problems with 'spoiled' princes, but our sources depict this more in line with alcoholism and greed than growing up with fine things. It is notable that the maxims of Chinggis Khan which survive in the sources often relate to limiting alcohol consumption.
In the Yuan Dynasty (the Mongol Khanate in China and Mongolia ruled by Kublai Khan and his successors) the ruling court in Dadu (Khanbaliq, modern Beijing) was rather infamous for hedonism, the final Yuan ruler in China often said to have spent his long 3 decade reign distracted by women, alcohol and effectively everything but ruling, though its been shown that this was exaggerated to a large degree by later Chinese writers, and efforts to contrast them with the new Ming Dynastic founder, the Hongwu Emperor. Hongwu wanted to be shown as just, firm, not distracted by worldly distractions as the final Yuan rulers were. We also see throughout the Yuan Dynasty conflict between whether the Mongols should maintain their steppe customs and focus of the empire, using China's wealth for expanding the empire -associated with hunting, herding, livings in gers and the military- or adopt sedentary customs and focus on ruling the Chinese -associated with living in palaces, living in China. This was a serious conflict which often turned to armed violence and coups, and few Yuan emperors died natural deaths.
Iggulden's (if this was in fact from him, I do apologize if this is wrong, but I can't access my copies of his books right now) depiction of Mongol reactions to children spoiled by sedentary (generally, Chinese) customs does in fact reflect a serious problem which the Mongols faced being a softer life with sedentary aspects, or the leaner world of the steppe. But hunting as punishment seems unlikely. When we are in the period of Mongols being ''returning' from their campaigns to Mongolia (so, the period from 1209-1260), those sons were in all possibility on campaign alongside their fathers (conscription for military service started around 15), or taking roles in the maintenance of the herds and production of arms and armour for the warriors, which even with the influx of slaves and goods from the conquests still needed to be done. If anything, taking a year off for hunting would be a reward!
Sources:
Morris Rossabi, Kublai Khan: His Life and Times
Timothy May, The Mongol Empire (2018)
Christopher Atwood, "Jochi and the Early Campaigns,"
Frederick W. Mote, "The Rise of the Ming Dynasty, 1330-1367," in the Cambridge History of China, Vol 7, The Ming Dynasty 1368-1644, Part one.
The Secret History of the Mongols, translated by Igor de Rachewiltz: https://cedar.wwu.edu/cgi/viewcontent.cgi?referer=https://www.youtube.com/&httpsredir=1&article=1003&context=cedarbooks