What does Buddhism teach about healing others through the mind

What happens when we aren't attentively listening, and then we see someone who attentively tries to listen to something?

It's natural if we would begin to listen attentively too.

So our behavior and states of mind influence others, and are influenced by others. Therefore, having clear and calm mind can benefit others around us.

It happened that entering a room or a Dharma center where a Buddhist teacher experienced in meditation was present, I felt thick atmosphere of awareness. Like a field of energy.

Likewise, sometimes I felt similar "field of energy" entering a church, or entering my room when I practiced there a lot.

There are many stories how a state of mind influenced others or even influenced weather, e.g. stopped a drought. Famous Heart Sutra (except in the brief version) says:

Once the Blessed One was dwelling in Rajagriha at Vulture Peak mountain, together with a great gathering of the sangha of monks and a great gathering of the sangha of bodhisattvas. At that time the Blessed One entered the samadhi that expresses the dharma called “profound illumination,” and at the same time noble Avalokiteshvara, the bodhisattva mahasattva, while practicing the profound prajñaparamita, saw in this way: he saw the five skandhas to be empty of nature.

It means the Buddha was in deep meditative absorption, and that helped Avalokiteshvara to immerse in deep clarity and gain insight.

I noticed such things in my own life. For example, when I sat down with a silent mind near people whose mind was scattered and troubled, often they realized something, e.g. solutions to their problems, or they recalled something they wanted to do.

Therefore we can say that harmony, clarity, calmness and concentration of our mind might help others too be in harmony, clarity, calmness and concentration.

When turbulence of mind ceases, our bodies naturally tend to optimize their self-regulation.

So there are Buddhist practices, especially in Mahayana, such as Bhaishajya Guru (https://en.wikipedia.org/wiki/Bhaisajyaguru) sadhanas, which are used for healing.

Among the mental factors conductive to awakening, there are Four bases of spiritual power, or riddhipada. See Iddhipada (Wikipedia) and Four Steps to Magical Powers by Master Sheng Yen.

Improving our states of mind as means of practice is the basis of the concept of Six paramitas. It might be considered the central concept of Mahayana Buddhism, with the role similar to the role of the Eightfold path for Buddhism in general.

Six paramitas (six perfections) can be explained as developing puṇya (Pali puñña, English merit) - the energy of good karma. As we practice the first paramita, selfless giving, and so on, we move the burden of egoism away from our shoulders. It can be felt as natural "unconditional happiness". Like feeling warm sun on the skin, warm smile inside, clarity and calmness pervading everything, as unlimited as space.

That process of abandoning egoism therefore is explained as developing the energy of good karma. Our actions become more powerful and clean, our mind more stable and perceptive, and so on.

When we try to study Dharma intellectually, we can find it's hard to understand complex ideas like emptiness etc., but when we practice Six perfections, we develop punya, and understanding becomes easier. Therefore Mahayana teachers say: "The path of giving is the supreme path". (See e.g. "Ratna avali").

Some students misunderstand that, theoretically thinking that such words were addressed to laymen and so weren't ultimately true. They think that it's clear that the supreme path is studying emptiness. But in real practice we develop punya, and that enables us to realize emptiness, so it's the same path.

Therefore we shouldn't think that developing good karmic energy is somehow a lower path; it is indeed the supreme path, inseparable from the realization of emptiness.

That's why Mahayana practitioners, especially after a session of formal practice, pronounce "dedication of merit". In order to keep our giving selfless, and to keep our energy from dissipation, we give it away, to benefit everybody. We dedicate the developed punya to the awakening of all beings, or to other goals in that direction. For example, after a session of meditation etc., we can transfer merit dedicating it to healing a sick person.

Also note that causes created mentally become more powerful when they are expressed in actions of body and speech.