Short answer - your secondary air pump (SAP) would not be able to pump air into the exhaust manifold likely popping a code.
Long answer - the one-way valve (black/white), the solenoid that is connected to the one-way valve, and associated vacuum hoses are used to actuate the SAP injection control valve. It looks something like this and it's mounted to the exhaust manifold. (Note there are two different versions of this control valve so yours may or may not look like this picture.)
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If you connect the black/white one-way valve the wrong way, this control valve would not get the necessary vacuum signal to open, therefore even if your SAP does run, it wouldn't be able to inject air into the exhaust manifold.