Beyond their applications in organic light-emitting diodes (OLEDs), thermally activated delayed fluorescence (TADF) materials can also make good photonic markers. Time-gated measurement of their delayed emission enables "background-free" imaging in, for example, biological systems, because no naturally-occurring compounds exhibit such long-lived emission. Attaching a strongly-absorbing antenna, such as a phenylene ethynylene oligomer, to the TADF core would be of interest to increase their brightness as photonic markers. With this motivation, we study a sequence of TADF-oligomer conjugates with oligomers of varying length and show that, even when the absorption of the oligomer is almost resonant with the charge-transfer absorption of the TADF core, the antenna transfers energy to the TADF core. We study this series of compounds with time resolved emission and transient absorption spectroscopy and find that the delayed fluorescence is essentially turned-off for the longer antennae. Interestingly, we find that the turn-off of the delayed fluorescence is not caused by quenching of the TADF charge-transfer triplet state due to triplet energy transfer of the lower-lying triplet state to the antenna, but must be associated with a decrease in the reverse intersystem crossing rate. These results are of relevance for the further development of TADF "dyes" and also, in the broader context, for understanding the dynamics of TADF molecules in the vicinity of energy donors/acceptors (i.e., in fluorescent OLEDs wherein TADF molecules are used as an assistant dopant).