I only vaguely recall the valvetrain setup, but you do have a VTEC
engine... there could be a valvetrain problem with the VTEC lifter
components which only makes noise as the ECM is adjusting valve lift
with the oil pressure control solenoid. Perhaps the rocker shafts are
worn where the VTEC rockers operate. If this were deemed by an
experienced ear to be likely, a simple removal of the valve cover would
reveal the shafts and rockers for some rudimentary inspection. For
that matter, the regular rockers could be worn also. It is hard to
diagnose from a static, written description.

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