To determine which action will most directly increase the magnetic flux in a magnetic core with a fixed cross-sectional area, we need to consider how magnetic flux is affected by different factors related to the core and magnetic field. The magnetic flux \(\Phi\) through a core is given by the formula:

\(\Phi = B \times A\)

where \(B\) is the magnetic flux density and \(A\) is the cross-sectional area of the core.

Since the area \(A\) is constant, we focus on increasing the magnetic flux density \(B\). According to Ampere's Law and the concept of magnetomotive force (MMF), \(B\) can be enhanced by increasing the product of the current and the number of turns in the coil:

\(B = \frac{\text{MMF}}{l}\)

where MMF = \((N \times I)\), \(N\) is the number of turns, \(I\) is the current, and \(l\) is the magnetic path length.

1. Using a material with higher resistance: Increasing resistance does not directly increase magnetic flux; instead, it may reduce current if the voltage is constant, thus not helping in increasing the flux.
2. Increasing the air gap in the magnetic path: This action would increase the reluctance of the core and actually decrease the magnetic flux.
3. Decreasing the magnetizing current: Reducing current would result in a lower MMF; hence, it would decrease the flux.
4. Increasing the number of turns or current producing the magnetomotive force: By increasing either the number of turns \(N\) or the current \(I\), the MMF increases, thereby increasing the magnetic flux.

Thus, the most direct way to increase the magnetic flux is option 4: Increasing the number of turns or current producing the magnetomotive force.