I had this same problem and ended up changing the ' Web Event Recording Configuration ' to record the onclick functions for WebElements.

1. Access the Web Event Recording Configuration dialog (From QTP main menu ''Tools'' > ''Web Event Recording Configuration'' option.
2. Click on ''Custom Settings...'' button to show ''Custom Web Event Recording Configuration'' dialog.
**Note:** On the left side, a tree view control shows different types of web test objects and on right side are shown all events being listened/recorded

3. Before moving forward, backup/export to a file the current settings, by going to ''File'' and selecting ''Save Configuration As''.
**Note:** exported file can be ''imported'' by going to ''File'' and selecting ''Load Configuration''

4. Expand ''Web Objects'' (under ''Any Web Object'' item on left side) and find/highlight/select test object showing problematic behavior (WebElement in this case).
5. Once selected a test object, add onclick event to the webelement. Object > Add > onclick.
6. Click ''Ok'' on currently opened dialogs to confirm changes and go to back to main screen of QTP
7. Confirm that this change allows you to record.