A StandaloneInputModule that only shows the mouse cursor if the mouse is active.

StandaloneInputModuleEx.cs

using UnityEngine;
using UnityEngine.EventSystems;

/// <summary>
/// A StandaloneInputModule replacement that only shows the mouse cursor if the mouse is active.
/// If keyboard or controller input is detected, the cursor is disabled. If mouse input is
/// detected, the cursor is enabled.
/// 
/// This also has an optional setting to disable clearing the input focus when clicking on
/// an empty part of the screen.
/// </summary>
public class StandaloneInputModuleEx : StandaloneInputModule
{
    [Tooltip("If true, clear the current focus when clicking on an empty part of the screen.")]
    [SerializeField]
    private bool m_clearFocusOnEmptyClick = true;

    #region Private state

    private Vector2 m_mousePosition;
    private Vector2 m_lastMousePosition;
    private bool m_cursorVisible;

    #endregion

    #region Public interface

    public override bool ShouldActivateModule()
    {
        // The keyboard or controller are active if a button was pressed or an axis changed.
        var keyboardIsActive = Input.GetButtonDown(submitButton);
        keyboardIsActive |= Input.GetButtonDown(cancelButton);
        keyboardIsActive |= !Mathf.Approximately(Input.GetAxisRaw(horizontalAxis), 0.0f);
        keyboardIsActive |= !Mathf.Approximately(Input.GetAxisRaw(verticalAxis), 0.0f);

        // The mouse is active if the first button went down or it was moved.
        m_lastMousePosition = m_mousePosition;
        m_mousePosition = Input.mousePosition;
        var delta = m_mousePosition - m_lastMousePosition;
        var mouseIsActive = delta.sqrMagnitude > 0.0f;
        mouseIsActive |= Input.GetMouseButtonDown(0);

        if (mouseIsActive)
        {
            Cursor.visible = true;
        }
        else if (keyboardIsActive)
        {
            Cursor.visible = false;
        }

        return base.ShouldActivateModule();
    }

    public override void Process()
    {
        var usedEvent = SendUpdateEventToSelectedObject();

        if (eventSystem.sendNavigationEvents)
        {
            if (!usedEvent)
                usedEvent |= SendMoveEventToSelectedObject();

            if (!usedEvent)
                SendSubmitEventToSelectedObject();
        }

        if (m_cursorVisible != Cursor.visible)
        {
            // Cursor visibility has changed.
            m_cursorVisible = Cursor.visible;

            if (!Cursor.visible)
            {
                // Cursor is not visible, so clear the current selection to remove any
                // mouse highlight. Unfortunately this also clears the selected game
                // so we need to restore the selection afterwards.
                var selected = eventSystem.currentSelectedGameObject;
                ClearSelection();
                eventSystem.SetSelectedGameObject(selected);
            }
        }

        if (Cursor.visible)
        {
            var process = true;
            if (!m_clearFocusOnEmptyClick)
            {
                // If the mouse is not hovering over anything and a mouse button is pressed or released,
                // don't process mouse events to avoid clearing the current selection.
                var mouseData = GetMousePointerEventData();
                var buttonState = mouseData.GetButtonState(PointerEventData.InputButton.Left);
                var leftButtonData = buttonState.eventData.buttonData;
                var hoverObject = leftButtonData.pointerCurrentRaycast.gameObject;
                if (hoverObject == null && (mouseData.AnyPressesThisFrame() || mouseData.AnyReleasesThisFrame()))
                    process = false;
            }

            if (process)
                ProcessMouseEvent();
        }
    }

    #endregion
}