<p>We extend the recently discovered phenomenon of hidden zeros to tree amplitudes for Yang-Mills (YM) and general relativity (GR) theories with higher-derivative interactions. This includes gluon amplitudes with a single insertion of the local <i>F</i>³ operator, as well as graviton amplitudes at sub-leading and sub-sub-leading orders in the low-energy expansion of bosonic closed string amplitudes — referred to as <i>R</i>² and <i>R</i>³ amplitudes, respectively. The kinematic condition for hidden zeros leads to unavoidable propagator singularities in unordered graviton amplitudes. We investigate in detail the systematic cancellation of these divergences, which resolves ambiguities in the proof of hidden zeros. Our approach is based on universal expansions that express tree amplitudes as linear combinations of bi-adjoint scalar amplitudes.</p>