It will take a multi-stemmed approach (sorry I had to - pun intended) :)
Seriously though, you will need to get initial control through cutting the top of the woody invasive, effectively top killing the honeysuckle. You will need to cut and recut many times. You will need to establish or re-establish native plants for competition and for fuel to perform future controlled burns. You will need to perform controlled burns. The process described below will work on just about any invasive woody species.
If you have a remnant woodland, you may consider letting it heal on its own if it has the potential to regrow from the “seed bank”. However, many woodlands are so degraded that they need some help. In that case, install an appropriate seed mix to establish a diverse native community. Plug (planting small live plants) in appropriate plants to augment the ecosystem. Honeysuckle is commonly found in woodland so this woodland seed mix may work for you. Also be sure to use warrior plants to get your best competition. We would love to include Carex pensylvanica (Common Oak Sedge or Pennsylvania sedge) in the seed mix however, it does not typically produce enough seed for us to sell commercially, so it is best installed by plugs.
At minimum two times per season, you will need to weed whack/string trim resprouts when they are tender, before they become woody. This is typically done in May and again in September, but it will depend on your site and conditions that season.
The most important step with weed whacking it to not just cut resprouts, but also to whack the stem to damage any of the nodes where future stems could emerge, and to expose the stump to more damage allowing disease to enter and wasting much more of its energy repairing the area. Whack the resprouts then whack fully around the circumference of the stem to get this effect.
Controlled burning can be extremely effective as it top kills the woody invasive all the way to the ground, including injuring its ability to resprout. Although very effective controlled burns are not perfect. After a burn, wait until the next spring and evaluate areas or individual honeysuckle that were not top killed by the fire. Then perform a “cut and drop”, cutting any honeysuckle not controlled by the fire. You can also us a propane torch to target small scale invasions to gain even better control.
Let the fire do the work and only expend valuable human resourced on what the fire did not control. This two-pronged approach is the single best tool against any invasive woody species.
Remember to get nature on your side. Identify any areas lacking in native plant populations and continue to augment as mentioned above.
Any of these techniques can work on their own, but when they are used in combination, they are significantly more effective. This example is one linear iteration of Flywheel Ecology. When you repeat the steps above over time, time and time again, the ecosystem builds momentum, and it will unstoppable. Discipline is critical to achieving success, please commit to many seasons of this work to gain substantial control.