Since you already set up a flow that retrieves the SharePoint item using the Client Name filter, ensure the GET ITEMS action query is setup correctly (the quotation marks are necessary as well):
ClientName eq 'PowerAppsParameterHere'
Then add another step in your flow to "Respond to a PowerApp" and return all the relevant fields as separate outputs.
Example, ClientName as Text(String) and use the output from the GET ITEMS as the value
Then call the flow from your powerapp and use a variable to store the response (be sure to add the flow into your app first):
Example: Set(varClientData, YourFLowName.Run(txtClientName.Text)
Then update the Default properties of your form fields to pull values from varClientData:
Example:
If(chkStartFromExisting.Value, varClientData.Title,"")
I hope that's enough to get you headed in the right direction.