React - clearing an input value after form submit

Since you input field is a controlled element, you cannot directly change the input field value without modifying the state.

Also in

onHandleSubmit(e) {
    e.preventDefault();
    const city = this.state.city;
    this.props.onSearchTermChange(city);
    this.mainInput.value = "";
  }

this.mainInput doesn't refer the input since mainInput is an id, you need to specify a ref to the input

<input
      ref={(ref) => this.mainInput= ref}
      onChange={this.onHandleChange}
      placeholder="Get current weather..."
      value={this.state.city}
      type="text"
    />

In you current state the best way is to clear the state to clear the input value

onHandleSubmit(e) {
    e.preventDefault();
    const city = this.state.city;
    this.props.onSearchTermChange(city);
    this.setState({city: ""});
  }

However if you still for some reason want to keep the value in state even if the form is submitted, you would rather make the input uncontrolled

<input
      id="mainInput"
      onChange={this.onHandleChange}
      placeholder="Get current weather..."
      type="text"
    />

and update the value in state onChange and onSubmit clear the input using ref

 onHandleChange(e) {
    this.setState({
      city: e.target.value
    });
  }

  onHandleSubmit(e) {
    e.preventDefault();
    const city = this.state.city;
    this.props.onSearchTermChange(city);
    this.mainInput.value = "";
  }

Having Said that, I don't see any point in keeping the state unchanged, so the first option should be the way to go.

You are having a controlled component where input value is determined by this.state.city. So once you submit you have to clear your state which will clear your input automatically.

onHandleSubmit(e) {
    e.preventDefault();
    const city = this.state.city;
    this.props.onSearchTermChange(city);
    this.setState({
      city: ''
    });
}

this.mainInput doesn't actually point to anything. Since you are using a controlled component (i.e. the value of the input is obtained from state) you can set this.state.city to null:

onHandleSubmit(e) {
  e.preventDefault();
  const city = this.state.city;
  this.props.onSearchTermChange(city);
  this.setState({ city: '' });
}