Here are a couple of examples from the Renaissance and not the middle ages.

Edward VI 1537-1553) became king of England in 1547 age nine under a regency. Henry VIII's will established a council of regents until Edward VI reached the age of 18.

https://en.wikipedia.org/wiki/Edward_VI_of_England#Council_of_Regency1

Since Edward died aged 15 years, 8 months, and 24 days, he never achieved his legal majority and the regency council had the legal right to make decisions for him during his entire reign.

Edward VI was intelligent and well educated, and historians have speculated how much the regency council considered his opinions and whether Edward would reverse their decisions when he came of age.

As Edward was growing up, he was able to understand more and more government business. However, his actual involvement in decisions has long been a matter of debate, and during the 20th century, historians have presented the whole gamut of possibilities, "balanc[ing] an articulate puppet against a mature, precocious, and essentially adult king", in the words of Stephen Alford.[104] A special "Counsel for the Estate" was created when Edward was fourteen. Edward chose the members himself.[105] In the weekly meetings with this Council, Edward was "to hear the debating of things of most importance".[106] A major point of contact with the king was the Privy Chamber, and there Edward worked closely with William Cecil and William Petre, the Principal Secretaries.[107] The king's greatest influence was in matters of religion, where the Council followed the strongly Protestant policy that Edward favoured.[108]

https://en.wikipedia.org/wiki/Edward_VI_of_England#Council_of_Regency1

I once read a biography of Edward VI that stated that when Edward wanted to forbid his half sister Mary to have catholic worship, Mary's cousin Emperor Charles V threatened to invade England with a Spanish and Dutch armada and conquer it unless Edward agreed to let Mary worship as she wished. The biography said that it took the council members hours of arguing and pleading with Edward that they had to save England from conquest by letting Mary have Catholic worship before Edward agreed to it.

I also read somewhere that at one Christmas celebration Edward stated that the ceremonies seemed superstitious, and that the reforming group took that as grounds to proceed swiftly with making the Church of England more Protestant. And the writer of that account claims that shows that Edward VI was actually the most absolute monarch in English history, even more so than his father Henry VIII.

When Edward VI was dying in 1553, the government decreed that his half sisters Mary and Elizabeth were not eligible to inherit the throne, and that the throne should go to their cousin Lady Jane Gray.

For centuries, the attempt to alter the succession was mostly seen as a one-man-plot by the Duke of Northumberland.[165] Since the 1970s, however, many historians have attributed the inception of the "devise" and the insistence on its implementation to the king's initiative.[166] Diarmaid MacCulloch has made out Edward's "teenage dreams of founding an evangelical realm of Christ",[167] while David Starkey has stated that "Edward had a couple of co-operators, but the driving will was his".[168] Among other members of the Privy Chamber, Northumberland's intimate Sir John Gates has been suspected of suggesting to Edward to change his devise so that Lady Jane Grey herself—not just any sons of hers—could inherit the Crown.[169] Whatever the degree of his contribution, Edward was convinced that his word was law[170] and fully endorsed disinheriting his half-sisters: "barring Mary from the succession was a cause in which the young King believed."[171]

https://en.wikipedia.org/wiki/Edward_VI_of_England#Devise_for_the_succession2

So even though Edward VI never reached his legal age of majority, I can remember three accounts of Edward's will and actions allegedly changing the course of English history. If those accounts are correct the regency council apparently feared punishment after Edward reached his majority and took power if they took actions that he strongly opposed.

King Francis II of France (19 January 1544-5 December 1560) lived to be 16 years, 10 months, and 16 days old. He became King of France on 10 July 1559 aged 15 years, 5 months, and 16 days old.

According to French law, Francis at the age of fifteen was an adult who in theory did not need a regent.[5] But since he was young, inexperienced, and in fragile health, he delegated his power to his wife's uncles from the noble House of Guise: Francis, Duke of Guise, and Charles, Cardinal of Lorraine. His mother, Catherine de' Medici, agreed to this delegation. On the first day of his reign, Francis II instructed his four ministers to take orders from his mother, but since she was still in mourning for the loss of her husband, she directed them to the House of Guise.[6]

https://en.wikipedia.org/wiki/Francis_II_of_France3

As far as I know, Francis II probably simply rubber stamped the decisions made by his mother Catherine and the Guises. If Francis did personally decree anything, it probably would have been instigated by his strong willed wife, Mary Queen of Scots.

So the differences between the reigns of Edward VI and Francis II are in large part due to how different their personalities were.

And in the Renaissance society and government was similar to in the later Middle Ages, so there should have been similar variations during the Middle Ages.

Very broad question in both time and space. I'm going to offer a handwave summary, and then point out some research avenues.

OP appears to make the assumption that when the monarch died, the child would become the monarch -this is not a universal assumption. To my mind this restricts the question to the later medieval period; that may help to refine the question. (My impression is that succession protocol was less well defined in the early Medieval period; I associate automatic succession with the late Medieval period. I don't have the time to support that impression right now, and I'm open to correction).

Handwave summary

Most monarchies are an explosive mixture of competing internal factions; if the monarchy is not secured by a combination of military power, political skill, and governance institutions, the monarchy will descend (or explode) into civil war. The child king is just a tool to add legitimacy to a regency; the child's wishes are not particularly relevant to the job. If the alignment of military/political power is strong enough to preserve the country until the child comes of age, the only qualification that the child needs is the potential to eventually produce an heir and a spare.

I am strongly skeptical that the child's opinion would have any influence on the regent or the regency council. The primary purpose of the Regent/regency is to preserve the country until the ruler comes of age. The regent needs to have military/political power required for that task.

OP asked how much power the child monarch could have; as much as the limits permit. If the child monarch is particularly skillful at politics, the child monarch may be able to play the factions against one another. The child monarch is unlikely to have any military power, and aligning with any existing faction will effectively make the child monarch subservient to the faction. The child monarch's only successful tactic is to play the game that the King of France did -to ensure that all non-royal factions are equally weak. I suppose it is possible that there might be a faction that is based on the legitimacy of the monarch, and willing to back the monarch without attempting to control the monarch. That is unlikely (the iron law of power is that the survival of the faction is more important than the justification for the faction), but in such a position the child monarch might rule - of course in that situation the regent would very likely be from the royalist faction.

Further research

If I were to research further to develop case studies, I'd start with

• List of Child Rulers (Medieval child rulers)
• List of Child Monarchs who Changed History
• Regency

Examples

• @Luiz offers that, "the child-king may be completely or almost powerless until maturity. e.g. D. Pedro II of Brazil (XIX century) and the movement for declaring his early maturity. Including this here because answers should be in answers, not in comments; full credit to @Luiz.

The obvious complexity is that (just to pick an example) Spanish monarchy is not like English monarchy, which is not like French monarchy, which is distinct from German monarchy, etc. (OP limits the question to Europe, so I don't have to consider Japanese or monarchy, etc.)

Also complicating the question is that in this case, history is local. The answer will depend strongly on the personalities of the monarch, the age of the monarch, the power dynamics of the country, and the presence or absence of governance institutions. The situation will be quite different in Anglo Saxon England than in England.