Do any Christian groups or denominations believe that miracle-working faith can be cultivated through prayer and fasting?

The Church of Jesus Christ of Latter-day Saints believe that fasting...:

Fasting can have many purposes. We can overcome weaknesses or problems by fasting and praying. Sometimes we may wish to fast and pray for help or guidance for others, such as a family member who is ill and needs a blessing (see Mosiah 27:22–23)....

We can fast to help others embrace the truth. Fasting can help comfort us in times of sorrow and mourning (see Alma 28:4–6). Fasting can help us become humble and feel closer to our Heavenly Father (see Helaman 3:35)....

When we fast wisely and prayerfully, we develop our faith. With that faith we will have greater spiritual power.

.. can and does bring about miracles:

A miracle is an extraordinary event caused by the power of God. Miracles are an important element in the work of Jesus Christ. They include healings, restoring the dead to life, and resurrection. Miracles are a part of the gospel of Jesus Christ. Faith is necessary in order for miracles to be manifested

Some examples & reading on miracles/fasting

• Is Not This the Fast I Have Chosen, Pres Henry B Eyring
• Miracles, Elder Dallin H Oaks
• Fasting - A Gift of Joy, Sheryl Condie Kempton
• Blessings of Fasting, Mindy Raye Friedman
• The Blessings of a Proper Fast, Carl B Pratt
• Miracles, Matthew Cowley
• 7 Experiences from Worldwide Fast Participants You Don't Want to Miss (less of an official source, but has some good examples found on social media)

_{All emphasis mine}

@KenGraham's answer makes some good points on which I will elaborate.

Quoting Ken:

Thus it is obvious that both prayer and fasting (penance) must be done in conjunction with Christian faith, piety and purity of living. We must avoid being white washed tombs of the Pharisees while praying or fasting.

I will use the analogy of an illness. An internal medicine specialist will observe symptoms, diagnose an illness, and prescribe the right medicine. A nutritionist will check for vitamin or other nutritional deficiencies, test for malabsorption of nutrients, and recommend changes to diet. You can carry this on to other specialties. Sometimes the problem has but one treatment, other times there is overlap in which several alternatives will work equally well or complement one another to speed recovery.

In the passage about certain demons coming out only by prayer and fasting, was Jesus recommending the only solution for a particular case, like the pharmacological approach where you need exactly the right drug? Or was he observing a general spiritual "nutritional" deficiency in the life and practice of the disciples that needed to be remedied so that they could have a healthy spiritual life?

In studying Job, I discovered thirteen different means of communication with God that Job had access to and employed during his life. I believe that pursuing all such means as part of a well-balanced spiritual life is the best course. Prayer and fasting were among the means Job used. So were meditating on God's words (such as were available in his day), trusting the words of wise elders, observing God through nature, learning from suffering, being blessed by miracles and messages from angels and several more. Chief among them was hearing words straight from God's mouth via theophany. Job's faith was holistic. He maintained wide open communication with God in every way he knew how, and God blessed him with miraculous interventions that expanded and deepended that exchange.

The passage about "the faith of a mustard seed" can give the wrong impression. Another parable is instructive, the one where a farmer plants his seeds and waits for the plant to grow, but he doesn't know how it grows, it just does. The important ingredient is time. Do not worry about trying to grow your faith. Spend time in communion with the Lord and the growth in faith will happen naturally.

I once prayed and fasted weekly for three years. I prayed for the same thing each of those weeks. I finally stopped because I just knew that God had heard me. A year or so after I stopped my fast, I began to receive what I had prayed for and continue to receive it to this day. My prayer? I had read of Moses' bold request to the Lord to show him His glory. So I prayed, "Lord, show me your glory." I left it up to God in what form that would come. I saw no glowing apparition pass by, but I did see God's glory in His word. In the four or five years since then, my understanding of the Bible has blossomed. Passages that once mystified me now speak to me clearly.

How can I be sure it was the fasting that did this? For twenty years my understanding of the Bible had stagnated or grown slowly, requiring great effort to acquire small returns. That fast was the only major change I made to my religious practice during that time. It was either that or pure grace completely unrelated to my actions. More likely it was God's pure grace that caused me to start fasting in the first place, leading to the desired outcome.

James was right. I do not generally pray for more faith, though I should like more faith. Instead I assume that I have enough faith for the task, however large. All I need to do is pray consistently, fast semi-regularly, search Scripture for wisdom often, etc. Immediate results occasionally come, but sometimes weeks, months or years of prayer are required. It would seem that the delay in receiving an answer is an indication of weak faith, but it may be the other way around. A lengthy delay in receiving what we pray for - when we pray regulary for it, without ceasing and do not surrender hope - indicates a strong faith, not a weak one.

Do any Christian groups or denominations believe that miracle-working faith can be cultivated through prayer and fasting?

Generally speaking, many Christians believe in this, regardless of denomination. However, Our Lord warns us that we must be careful not to pray or fast as the Pharisees do.

Now John’s disciples and the Pharisees were often fasting. So people came to Jesus and asked, “Why don’t Your disciples fast like John’s disciples and those of the Pharisees?” - Mark 2:18

When you fast, do not be somber like the hypocrites, for they disfigure their faces to show men they are fasting. Truly I tell you, they already have their full reward. But when you fast, anoint your head and wash your face... - Matthew 6:16

And when you pray, do not be like the hypocrites. For they love to pray standing in the synagogues and on the street corners to be seen by men. Truly I tell you, they already have their full reward. But when you pray, go into your inner room, shut your door, and pray to your Father, who is unseen. And your Father, who sees what is done in secret, will reward you. - Matthew 6:5

Thus it is obvious that both prayer and fasting (penance) must be done in conjunction with Christian faith, piety and purity of living. We must avoid being white washed tombs of the Pharisees while praying or fasting.

An excellent example of obtaining favours (miracles) from God, through prayer and fasting is in the famous Curé D'Ars. St. John Vianney was well known for his extraordinary prayer life and fasting by which he was able to operate many miracles both during his lifetime and after his death.

This humble Catholic priest explains this in his own words as follows:

If you are seeking a miracle, see no way out of a situation, or are on the verge of giving in to the temptation of despair - please do not lose hope! If your health permits, you may wish to consider offering a fast along with the prayers of this novena.

Here is what the holy, humble Curé himself said about the efficacy of fasting:

Regarding mortification, he once said, “My friend, the devil is not greatly afraid of the discipline and other instruments of penance. That which beats him is the curtailment of one’s food, drink and sleep. There is nothing the devil fears more, consequently, nothing is more pleasing to God. Oh! How often have I experienced it! Whilst I was alone – and I was alone during eight or nine years, and therefore quite free to yield to my attraction – it happened at times that I refrained from food for entire days. On those occasions I obtained, both for myself and for others, whatsoever I asked of Almighty God.” - Curé of Ars: The Importance of Fasting

St. Jean-Baptiste-Marie Vianney (8 May 1786 – 4 August 1859), is venerated as a Catholic saint. He was a French Catholic priest who is now the patron saint of parish priests. He is often referred to as the "Curé d'Ars" or the parish priest of Ars.

The Devil told St. John Vianney: "If there were three such priests as you, my kingdom would be ruined."

According to Father Trochu’s book (from a deposition taken from Father Alfred Monin, a young priest), John Vianney was in the church hearing confessions when he was informed of the fire in his room. “The Grappin is very angry,” Vianney remarked. “He couldn’t catch the bird so he has burned the cage. It is a good sign. We will have many sinners this day.”

Many such saintly persons along with similar teachings can be found in various Christian denominations, especially amongst Catholic and Orthodox saints, the Desert Fathers and the founders of early monasticism.