What sort of problems did the Church face because of fedualism in the High Middle Ages?
The major problems of the church were not so much the result of feudal practice as they were corruption within the church itself. At a time when lands held by the nobility were typically passed to the eldest son, second and third sons were often tonsured and inducted into the upper clergy where they enjoyed the same benefits of a feudal lord, such as collection of taxes in kind, lands operated by serfs, etc. High church offices were often quite lucrative, and for that reason, were often the subject of intense competition. Those who achieved high office normally were there for personal gain rather than dedication to service. The office of Pope, easily the most lucrative position, was often bought and paid for as a political favor. A classic example was Pope John XII, elected to the office at age seventeen and the youngest man ever to hold the office. He only received it because his father was a cardinal and used his influence to have his son elected. Obviously, the father had broken his vows of celibacy; and his son was no exception. He kept a virtual harem within the Vatican, and died in his early twenties in the arms of a married woman.
Another sad example of corruption was Pope Alexander VI who drew the famous Line of Demarcation. Alexander raised two of his own children, Cesare and Lucretia Borgia in the Vatican and is rumored to have engaged in an incestuous relationship with his own daughter. Pope Martin V wrote to his brother after his election, "God has given us the papacy, let us enjoy it."
The only Pope to ever resign, Celestine V, did so after his successor, the future Bonface VIII, whispered into the Pope's chamber (he lived as a hermit) pretending to be the voice of God and urging him to resign. Boniface himself used the church to enrich himself and often referred to himself as Caesar.
The corruption of the church was its major problem. The various Popes did engage in vociferous battles with Emperors over secular vs. clerical control; yet these disputes were a much greater problem for the Emperors than for the Church.
The major problem that the Catholic Church faced because of feudalism was competition for control both of secular and church matters.
Feudal lords liked to have complete control of their domains. They had power over their serfs and the other peasants. They were the law. They also, therefore, came to want to control the Church. They wanted to have the ability to do things like appointing bishops. This led to conflict between the Church and the feudal lords.
There was also conflict over secular power. The Church still wanted to exercise power over secular lords. It felt that, as the representative of God, it should have power on Earth as well.