In English, an adjective can appear after a noun, but it has a different nuance -- more stylized, poetic, or dramatic -- from the more typical placement.
I walked through the woods alone filled with dark thoughts and with a heavy heart.
I walked through the woods alone, my thoughts dark and my heart heavy.
A "weak voice" is one which is quiet or uncertain, and which possibly stutters or falters in a way that suggests fear, doubt, or insincerity.
As Tᴚoɯɐuo mentions in his comment, these kind of expressions are called "absolute phrases" and don't always contain a verb, but do contain information relevant to the overall sentence. BillJ (see comment below) classifies it as "a verbless clause functioning as a supplement". It can be changed to a finite clause by adding the "to be" verb:
My voice was weak as I replied to the secretary. "Fine," I lied.