You need to trigger the onChange event manually. On text inputs onChange listens for input events.

So in you handleClick function you need to trigger event like

handleClick () {
    this.setState({value: 'another random text'})
    var event = new Event('input', { bubbles: true });
    this.myinput.dispatchEvent(event);
  }

Complete code

class App extends React.Component {
  constructor(props) {
    super(props)
    this.state = {
    value: 'random text'
    }
  }
  handleChange (e) {
    console.log('handle change called')
  }
  handleClick () {
    this.setState({value: 'another random text'})
    var event = new Event('input', { bubbles: true });
    this.myinput.dispatchEvent(event);
  }
  render () {
    return (
      <div>
        <input readOnly value={this.state.value} onChange={(e) => {this.handleChange(e)}} ref={(input)=> this.myinput = input}/>
        <button onClick={this.handleClick.bind(this)}>Change Input</button>
      </div>
    )
  }
}

ReactDOM.render(<App />,  document.getElementById('app'))

Codepen

Edit: As Suggested by @Samuel in the comments, a simpler way would be to call handleChange from handleClick if you don't need to the event object in handleChange like

handleClick () {
    this.setState({value: 'another random text'})
    this.handleChange();
  }

I hope this is what you need and it helps you.

I tried the other solutions and nothing worked. This is because of input logic in React.js has been changed. For detail, you can see this link: https://hustle.bizongo.in/simulate-react-on-change-on-controlled-components-baa336920e04.

In short, when we change the value of input by changing state and then dispatch a change event then React will register both the setState and the event and consider it a duplicate event and swallow it.

The solution is to call native value setter on input (See setNativeValue function in following code)

Example Code

import React, { Component } from 'react'
export class CustomInput extends Component {

    inputElement = null;
    
    // THIS FUNCTION CALLS NATIVE VALUE SETTER
    setNativeValue(element, value) {
        const valueSetter = Object.getOwnPropertyDescriptor(element, 'value').set;
        const prototype = Object.getPrototypeOf(element);
        const prototypeValueSetter = Object.getOwnPropertyDescriptor(prototype, 'value').set;

        if (valueSetter && valueSetter !== prototypeValueSetter) {
            prototypeValueSetter.call(element, value);
        } else {
            valueSetter.call(element, value);
        }
    }


    constructor(props) {
        super(props);

        this.state = {
            inputValue: this.props.value,
        };
    }

    addToInput = (valueToAdd) => {
        this.setNativeValue(this.inputElement, +this.state.inputValue + +valueToAdd);
        this.inputElement.dispatchEvent(new Event('input', { bubbles: true }));
    };

    handleChange = e => {
        console.log(e);
        this.setState({ inputValue: e.target.value });
        this.props.onChange(e);
    };

    render() {
        return (
            <div>
                <button type="button" onClick={() => this.addToInput(-1)}>-</button>
                <input
                    readOnly
                    ref={input => { this.inputElement = input }}
                    name={this.props.name}
                    value={this.state.inputValue}
                    onChange={this.handleChange}></input>
                <button type="button" onClick={() => this.addToInput(+1)}>+</button>
            </div>
        )
    }
}

export default CustomInput

Result

I think you should change that like so:

<input value={this.state.value} onChange={(e) => {this.handleChange(e)}}/>

That is in principle the same as onClick={this.handleClick.bind(this)} as you did on the button.

So if you want to call handleChange() when the button is clicked, than:

<button onClick={this.handleChange.bind(this)}>Change Input</button>

or

handleClick () {
  this.setState({value: 'another random text'});
  this.handleChange();
}