Linked Helper is not a cloud solution, it works locally on your PC and cannot perform any action when it is stopped or when the PC is turned off / in a sleeping state.
Until the invitation is accepted or the person replies anything to your invite, the invitation message is stored in Sent invites page: https://www.linkedin.com/mynetwork/invitation-manager/sent/.
Once the invite is accepted, a new message thread will appear in your Messaging page https://www.linkedin.com/messaging/ with the date referring to the moment of accepting the invitation, not the moment when the invitation was sent.
Note: If the recipient responds to your custom message, a new message thread will appear in your Inbox. A response can be sent without accepting your invitation: LinkedIn Personalizing Invitations to Connect.
If you do not want to receive the invitations sent previously, you need to withdraw them: How to cancel sent pending invites?