  Phase slips occur across all Josephson junctions (JJs) at a rate that
increases with the impedance of the junction. In superconducting qubits
composed of JJ-array superinductors -- such as fluxonium -- phase slips in the
array can lead to decoherence. In particular, phase-slip processes at the
individual array junctions can coherently interfere, each with an
Aharonov--Casher phase that depends on the offset charges of the array islands.
These coherent quantum phase slips (CQPS) perturbatively modify the qubit
frequency, and therefore charge noise on the array islands will lead to
dephasing. By varying the impedance of the array junctions, we design a set of
fluxonium qubits in which the expected phase-slip rate within the JJ-array
changes by several orders of magnitude. We characterize the coherence times of
these qubits and demonstrate that the scaling of CQPS-induced dephasing rates
agrees with our theoretical model. Furthermore, we perform noise spectroscopy
of two qubits in regimes dominated by either CQPS or flux noise. We find the
noise power spectrum associated with CQPS dephasing appears to be featureless
at low frequencies and not $1/f$. Numerical simulations indicate this behavior
is consistent with charge noise generated by charge-parity fluctuations within
the array. Our findings broadly inform JJ-array-design tradeoffs, relevant for
the numerous superconducting qubit designs employing JJ-array superinductors.
